Thank you, Alarik, for giving "form" to the unknown, by sharing the proper label given to that "beast".
After reviewing this article, I have to say that I am sitting on the fence, because I would rather not have bloat, but as I said elsewhere, we either let ourselves be carried away with the "currents" or we may drown from the attempt of staying put in the middle of the "onslaught". Very tough call! 
In case it is relevant as to the point of "breakdown", I am on UM 22.04.5 and the dconf-editor is showing with proper dark theme. So, if yours is forced to light at UM 24.04, then that is the definitely the transition point regarding "backward-compatibility" of GNOME apps, unless there was a Yaru theme "architecture tweak", which we don't know of.
You can find out what libraries an application uses with ldd. Must be a full path:
ldd $(which deja-dup)
You can see if uses GTK 4 or libadwaita - there might be other libraries.
The other side of the story is that it's by design - "stop theming their apps!" ... GNOME apps, anyway:
Déjà Dup would be a GNOME program. I think that's the trouble with MATE. GNOME do have stable, solid software, but it was designed for their platform guidelines. MATE would benefit from a separate "classic" frontend built in GTK, but that's kind of like reinventing the wheel.
Thank you, Luke, for that excellent reference to what I mentionned earlier on CSC (Client-Side Customization).
@lah7 and @ericmarceau Thank you for the insights.
Results of:
ldd $(which deja-dup)
linux-vdso.so.1 (0x00007fff8b7ee000)
libdeja.so => /usr/lib/x86_64-linux-gnu/deja-dup/libdeja.so (0x00007e1562e10000)
libadwaita-1.so.0 => /lib/x86_64-linux-gnu/libadwaita-1.so.0 (0x00007e1562a00000)
libgtk-4.so.1 => /lib/x86_64-linux-gnu/libgtk-4.so.1 (0x00007e1561e00000)
libpango-1.0.so.0 => /lib/x86_64-linux-gnu/libpango-1.0.so.0 (0x00007e1562d91000)
libgio-2.0.so.0 => /lib/x86_64-linux-gnu/libgio-2.0.so.0 (0x00007e1561c30000)
libgobject-2.0.so.0 => /lib/x86_64-linux-gnu/libgobject-2.0.so.0 (0x00007e1562d2c000)
libglib-2.0.so.0 => /lib/x86_64-linux-gnu/libglib-2.0.so.0 (0x00007e1561ae7000)
libpackagekit-glib2.so.18 => /lib/x86_64-linux-gnu/libpackagekit-glib2.so.18 (0x00007e1562cc9000)
libc.so.6 => /lib/x86_64-linux-gnu/libc.so.6 (0x00007e1561800000)
libgpg-error.so.0 => /lib/x86_64-linux-gnu/libgpg-error.so.0 (0x00007e1562ca4000)
libjson-glib-1.0.so.0 => /lib/x86_64-linux-gnu/libjson-glib-1.0.so.0 (0x00007e1562c78000)
libsecret-1.so.0 => /lib/x86_64-linux-gnu/libsecret-1.so.0 (0x00007e15629a1000)
libsoup-3.0.so.0 => /lib/x86_64-linux-gnu/libsoup-3.0.so.0 (0x00007e1561a4e000)
libfribidi.so.0 => /lib/x86_64-linux-gnu/libfribidi.so.0 (0x00007e1562c58000)
libgraphene-1.0.so.0 => /lib/x86_64-linux-gnu/libgraphene-1.0.so.0 (0x00007e1562983000)
libappstream.so.5 => /lib/x86_64-linux-gnu/libappstream.so.5 (0x00007e1561754000)
libm.so.6 => /lib/x86_64-linux-gnu/libm.so.6 (0x00007e156166b000)
libgmodule-2.0.so.0 => /lib/x86_64-linux-gnu/libgmodule-2.0.so.0 (0x00007e1562c4f000)
libpangocairo-1.0.so.0 => /lib/x86_64-linux-gnu/libpangocairo-1.0.so.0 (0x00007e1562c3e000)
libharfbuzz.so.0 => /lib/x86_64-linux-gnu/libharfbuzz.so.0 (0x00007e156155e000)
libcairo.so.2 => /lib/x86_64-linux-gnu/libcairo.so.2 (0x00007e156141a000)
libcairo-gobject.so.2 => /lib/x86_64-linux-gnu/libcairo-gobject.so.2 (0x00007e1562975000)
libfontconfig.so.1 => /lib/x86_64-linux-gnu/libfontconfig.so.1 (0x00007e15613c9000)
libgdk_pixbuf-2.0.so.0 => /lib/x86_64-linux-gnu/libgdk_pixbuf-2.0.so.0 (0x00007e1562947000)
libepoxy.so.0 => /lib/x86_64-linux-gnu/libepoxy.so.0 (0x00007e15612a6000)
libXi.so.6 => /lib/x86_64-linux-gnu/libXi.so.6 (0x00007e1561a3b000)
libX11.so.6 => /lib/x86_64-linux-gnu/libX11.so.6 (0x00007e1561169000)
libpangoft2-1.0.so.0 => /lib/x86_64-linux-gnu/libpangoft2-1.0.so.0 (0x00007e1561a20000)
libvulkan.so.1 => /lib/x86_64-linux-gnu/libvulkan.so.1 (0x00007e15610eb000)
libpng16.so.16 => /lib/x86_64-linux-gnu/libpng16.so.16 (0x00007e15610b3000)
libtiff.so.6 => /lib/x86_64-linux-gnu/libtiff.so.6 (0x00007e1561026000)
libjpeg.so.8 => /lib/x86_64-linux-gnu/libjpeg.so.8 (0x00007e1560fa3000)
libxkbcommon.so.0 => /lib/x86_64-linux-gnu/libxkbcommon.so.0 (0x00007e1560f5a000)
libwayland-client.so.0 => /lib/x86_64-linux-gnu/libwayland-client.so.0 (0x00007e1560f4a000)
libwayland-egl.so.1 => /lib/x86_64-linux-gnu/libwayland-egl.so.1 (0x00007e1561a1b000)
libXext.so.6 => /lib/x86_64-linux-gnu/libXext.so.6 (0x00007e1560f36000)
libXcursor.so.1 => /lib/x86_64-linux-gnu/libXcursor.so.1 (0x00007e1560f2a000)
libXdamage.so.1 => /lib/x86_64-linux-gnu/libXdamage.so.1 (0x00007e1561a16000)
libXfixes.so.3 => /lib/x86_64-linux-gnu/libXfixes.so.3 (0x00007e1560f22000)
libXrandr.so.2 => /lib/x86_64-linux-gnu/libXrandr.so.2 (0x00007e1560f15000)
libXinerama.so.1 => /lib/x86_64-linux-gnu/libXinerama.so.1 (0x00007e1560f10000)
libcairo-script-interpreter.so.2 => /lib/x86_64-linux-gnu/libcairo-script-interpreter.so.2 (0x00007e1560ee9000)
libthai.so.0 => /lib/x86_64-linux-gnu/libthai.so.0 (0x00007e1560ede000)
libz.so.1 => /lib/x86_64-linux-gnu/libz.so.1 (0x00007e1560ec2000)
libmount.so.1 => /lib/x86_64-linux-gnu/libmount.so.1 (0x00007e1560e75000)
libselinux.so.1 => /lib/x86_64-linux-gnu/libselinux.so.1 (0x00007e1560e48000)
libffi.so.8 => /lib/x86_64-linux-gnu/libffi.so.8 (0x00007e1560e3c000)
libpcre2-8.so.0 => /lib/x86_64-linux-gnu/libpcre2-8.so.0 (0x00007e1560da2000)
/lib64/ld-linux-x86-64.so.2 (0x00007e1562ef1000)
libgcrypt.so.20 => /lib/x86_64-linux-gnu/libgcrypt.so.20 (0x00007e1560c5a000)
libsqlite3.so.0 => /lib/x86_64-linux-gnu/libsqlite3.so.0 (0x00007e1560af0000)
libpsl.so.5 => /lib/x86_64-linux-gnu/libpsl.so.5 (0x00007e1560adc000)
libbrotlidec.so.1 => /lib/x86_64-linux-gnu/libbrotlidec.so.1 (0x00007e1560ace000)
libgssapi_krb5.so.2 => /lib/x86_64-linux-gnu/libgssapi_krb5.so.2 (0x00007e1560a7a000)
libnghttp2.so.14 => /lib/x86_64-linux-gnu/libnghttp2.so.14 (0x00007e1560a4f000)
libcurl-gnutls.so.4 => /lib/x86_64-linux-gnu/libcurl-gnutls.so.4 (0x00007e1560992000)
libxmlb.so.2 => /lib/x86_64-linux-gnu/libxmlb.so.2 (0x00007e1560968000)
libxml2.so.2 => /lib/x86_64-linux-gnu/libxml2.so.2 (0x00007e1560786000)
libyaml-0.so.2 => /lib/x86_64-linux-gnu/libyaml-0.so.2 (0x00007e1560765000)
libsystemd.so.0 => /lib/x86_64-linux-gnu/libsystemd.so.0 (0x00007e1560685000)
libzstd.so.1 => /lib/x86_64-linux-gnu/libzstd.so.1 (0x00007e15605cb000)
libstemmer.so.0d => /lib/x86_64-linux-gnu/libstemmer.so.0d (0x00007e15604fd000)
libfreetype.so.6 => /lib/x86_64-linux-gnu/libfreetype.so.6 (0x00007e1560431000)
libgraphite2.so.3 => /lib/x86_64-linux-gnu/libgraphite2.so.3 (0x00007e156040b000)
libXrender.so.1 => /lib/x86_64-linux-gnu/libXrender.so.1 (0x00007e15603ff000)
libxcb.so.1 => /lib/x86_64-linux-gnu/libxcb.so.1 (0x00007e15603d6000)
libxcb-render.so.0 => /lib/x86_64-linux-gnu/libxcb-render.so.0 (0x00007e15603c5000)
libxcb-shm.so.0 => /lib/x86_64-linux-gnu/libxcb-shm.so.0 (0x00007e15603c0000)
libpixman-1.so.0 => /lib/x86_64-linux-gnu/libpixman-1.so.0 (0x00007e1560311000)
libexpat.so.1 => /lib/x86_64-linux-gnu/libexpat.so.1 (0x00007e15602e5000)
libwebp.so.7 => /lib/x86_64-linux-gnu/libwebp.so.7 (0x00007e156026a000)
liblzma.so.5 => /lib/x86_64-linux-gnu/liblzma.so.5 (0x00007e1560236000)
libLerc.so.4 => /lib/x86_64-linux-gnu/libLerc.so.4 (0x00007e15601af000)
libjbig.so.0 => /lib/x86_64-linux-gnu/libjbig.so.0 (0x00007e156019f000)
libdeflate.so.0 => /lib/x86_64-linux-gnu/libdeflate.so.0 (0x00007e156018c000)
liblzo2.so.2 => /lib/x86_64-linux-gnu/liblzo2.so.2 (0x00007e156016b000)
libdatrie.so.1 => /lib/x86_64-linux-gnu/libdatrie.so.1 (0x00007e1560160000)
libblkid.so.1 => /lib/x86_64-linux-gnu/libblkid.so.1 (0x00007e1560125000)
libunistring.so.5 => /lib/x86_64-linux-gnu/libunistring.so.5 (0x00007e155ff78000)
libidn2.so.0 => /lib/x86_64-linux-gnu/libidn2.so.0 (0x00007e155ff56000)
libbrotlicommon.so.1 => /lib/x86_64-linux-gnu/libbrotlicommon.so.1 (0x00007e155ff33000)
libkrb5.so.3 => /lib/x86_64-linux-gnu/libkrb5.so.3 (0x00007e155fe68000)
libk5crypto.so.3 => /lib/x86_64-linux-gnu/libk5crypto.so.3 (0x00007e155fe3c000)
libcom_err.so.2 => /lib/x86_64-linux-gnu/libcom_err.so.2 (0x00007e155fe36000)
libkrb5support.so.0 => /lib/x86_64-linux-gnu/libkrb5support.so.0 (0x00007e155fe29000)
librtmp.so.1 => /lib/x86_64-linux-gnu/librtmp.so.1 (0x00007e155fe0b000)
libssh.so.4 => /lib/x86_64-linux-gnu/libssh.so.4 (0x00007e155fd98000)
libnettle.so.8 => /lib/x86_64-linux-gnu/libnettle.so.8 (0x00007e155fd43000)
libgnutls.so.30 => /lib/x86_64-linux-gnu/libgnutls.so.30 (0x00007e155fb49000)
libldap.so.2 => /lib/x86_64-linux-gnu/libldap.so.2 (0x00007e155faeb000)
liblber.so.2 => /lib/x86_64-linux-gnu/liblber.so.2 (0x00007e155fadb000)
libicuuc.so.74 => /lib/x86_64-linux-gnu/libicuuc.so.74 (0x00007e155f800000)
libcap.so.2 => /lib/x86_64-linux-gnu/libcap.so.2 (0x00007e155facc000)
liblz4.so.1 => /lib/x86_64-linux-gnu/liblz4.so.1 (0x00007e155faaa000)
libbz2.so.1.0 => /lib/x86_64-linux-gnu/libbz2.so.1.0 (0x00007e155fa96000)
libXau.so.6 => /lib/x86_64-linux-gnu/libXau.so.6 (0x00007e155fa90000)
libXdmcp.so.6 => /lib/x86_64-linux-gnu/libXdmcp.so.6 (0x00007e155fa86000)
libsharpyuv.so.0 => /lib/x86_64-linux-gnu/libsharpyuv.so.0 (0x00007e155fa7e000)
libstdc++.so.6 => /lib/x86_64-linux-gnu/libstdc++.so.6 (0x00007e155f400000)
libgcc_s.so.1 => /lib/x86_64-linux-gnu/libgcc_s.so.1 (0x00007e155fa50000)
libkeyutils.so.1 => /lib/x86_64-linux-gnu/libkeyutils.so.1 (0x00007e155fa49000)
libresolv.so.2 => /lib/x86_64-linux-gnu/libresolv.so.2 (0x00007e155fa34000)
libhogweed.so.6 => /lib/x86_64-linux-gnu/libhogweed.so.6 (0x00007e155f7b8000)
libgmp.so.10 => /lib/x86_64-linux-gnu/libgmp.so.10 (0x00007e155f734000)
libcrypto.so.3 => /lib/x86_64-linux-gnu/libcrypto.so.3 (0x00007e155ee00000)
libp11-kit.so.0 => /lib/x86_64-linux-gnu/libp11-kit.so.0 (0x00007e155ec5c000)
libtasn1.so.6 => /lib/x86_64-linux-gnu/libtasn1.so.6 (0x00007e155fa1c000)
libsasl2.so.2 => /lib/x86_64-linux-gnu/libsasl2.so.2 (0x00007e155f71a000)
libicudata.so.74 => /lib/x86_64-linux-gnu/libicudata.so.74 (0x00007e155ce00000)
libbsd.so.0 => /lib/x86_64-linux-gnu/libbsd.so.0 (0x00007e155f704000)
libmd.so.0 => /lib/x86_64-linux-gnu/libmd.so.0 (0x00007e155fa0d000)
So it is indeed a GTK4 app.
I completely understand that GNOME apps are intended for the GNOME DE and I too appreciate the article about the GNOME style and client side way of GTK4. Hopefully this can be resolved in MATE in the future. I thought I saw Linux Mint was trying to do this (i.e., have a common application platform for Cinnamon, MATE, and Xfce).
I do currently have Linux Mint MATE 22.1 running on my secondary laptop. I know I also set that with a dark theme (likely Mint-Y). Just out of curiosity, I installed Deja Dup to see if the theme responds on LM. It does yield the same result - light theme.
Thanks
Jaymo
Update : After a bit of thinking I have a workaround (for now).
I switched out the GTK4 apps for their Qt alternatives.
Except for a couple programs installed using the SNAP package manager, the KDE programs installed using APT fit in well and use the dark theme. Not only that, many have menu bars that also fit in well with the MATE Desktop "traditional" look.
I know this adds a significant number of packages to my install. However, given my system is not hurting on resources, this is not a problem.
Thanks
Jaymo
Ugh...
Look I usually try at least to be positive. If people want their desktop to look flat monotone and ugly that's fine, it's their desktop even though they're wrong. My problem comes in when they insist on making "updates" which force me to have their desktop on my hardware.
Yaru is ugly. I actively hate it. I've thought Ambiance a nice theme since it arrived in Ubuntu 10.04 and have done my best to continue using it as long as I can on my system. The last few releases have completely taken that option from me because increasingly things break when I try to use it.
Meanwhile even with Yaru my system continues to break consistency because of the encroachment of Gnome on my Mate desktop. I don't know what the answer going forward is but as my desktop continues to fragment I honestly find myself considering alternatives such as getting a Mac because I won't run Windows. And I hate Gnome's version of it more than I hate KDE. (Nothing personal if you like KDE but it just doesn't work for me on a fundamental level.) And no, this isn't some limp "threat" it's just me acknowledging that if I can't keep my desktop like I want it to be then I need to look at alternatives. And Gnome as instigator of this nonsense is at the literal bottom of my list.
For all this talk about Mate being "complete" it doesn't mean it needs to be dead. There can still be iterative improvements and optimizations that don't take away from the stability of its code. I'm not against improvements and have been looking forward to seeing us get Wayland for a while now. It'd stupidly breaking the interface that I have issues with.
As someone who lived through the LSB period and the whole rise of intercompatible system themes designed to make a consistent interface this whole thing where desktops are deliberately breaking the look and feel of user's systems to maintain some internal branding is weird to me. It feels really regressive. I don't understand how we could go from one to the other.
About gtk-4 showing light themes while everything else is dark:
I encountered the same some time ago and wrote this solution.
If the first tip doesn't work then the second one will:
Thanks @tkn . The first option didn't work for me. For whatever reason, I can set the color-scheme in dconf to 'prefer-dark'. Following a reboot or turning off the computer and on again, it reverts back to 'prefer-light' and won't keep my setting.
I have done the second though and it works on everything except the App Center and Firmware Updater (which I assume are SNAP packages written in the FLUTTER language). I added the following:
export GTK_THEME='Yaru-MATE-dark'
to my ~/.profile.
However, this broke some GTK4 applications visually. Thus, I ended up replacing several GNOME applications on my system with Qt applications from the KDE family. This worked really well and the general layout of the applications (title bar, menu bar) fit in well with the MATE style of maintaining a traditional desktop (in my opinion).
I wonder if, going forward, this could be a consideration in Ubuntu MATE releases to ship some Qt applications instead of GTK4 applications that don't fit in well visually. I understand, though, the downside being that there would be a considerable number of packages added to the installation for the Qt/KDE dependencies.
Thanks for all your help. I'm not a fan of GNOME either, and my hope is that desktops like MATE and Xfce can have some default applications that fit in better with the desktop aesthetic.
Best regards
Jaymo
Yes, I agree. There is a good chance that switching to Qt will give a lot less headaches and a better, more seamless visual fit then trying to keep up and play whack-a-mole with the highly instable ( = constantly changing & backwards incompatible) 'our-way-or-the-highway' disaster that GNOME/GTK has become.
Half the applications I use are KDE/Qt anyway and they look more MATE-native than GNOME ever will.