It seems to me that there used to be a tool to change the login screen somewhere in the preferences, and then, it was removed, and since at least the 21.04 release, if you want to change the login screen, you have to manually edit config files as described in this thread: https://ubuntu-mate.community/t/how-to-change-login-screen-background-in-ubuntu-mate-21-04/24384
Are things supposed to remain like this permanently?
In the system menu, System -> Administration -> Login Window allows you to easily select you login window background. However, you need to be aware of which greeter you are using. This setting is only used by a single greeter. I believe that one greeter is slick-greeter, but I can't say definitively since my Ubuntu MATE VM is using the arctica greeter. You can identify which greeter you are using with the following command:
lightdm --show-config
Look for the setting for greeter-session.
Err, what? Where is that supposed to be? I don't see any entry with that name. Not even in the "All" part of the main menu. Which version are you using? I'm on 22.10 now, and I didn't have the entry you're describing since before 21.04. (BTW, from using your command line cammond, it looks like I'm using arctica.)
this is only available in earlier versions of UM, not in 22.04
But you can, like I did, swapout the 'arctica-greeter' for the 'slick-greeter' to get the 'Login Window' option back. 
See this:
As you can see, I'm running 22.10, and this version has the "Login Window" option. I've had this VM for a very long time, and have regularly upgraded to each semi-annual release. So, it's entirely possible that this option is present because it was there in an earlier release and was retained through the upgrades. It is fully functional; if I click the menu item, the Login Window settings appears.
Yes, that probably explains it. I don't even have a submenu called "System" in the main menu anymore; only "Administration", "System Tools", and "Control Center". (I would have shown you, but for some reason, the standard screenshot tool doesn't work for me while the main menu is open.)
I had the same issue trying to use the Windows Alt-PrintScreen combo to capture a picture of the VM window. I still don't understand why that wouldn't work, since Windows shouldn't care what is going on inside the VM. I finally got the image I posted by using the VirtualBox screen capture function: View -> Take Screenshot...
@Raphael In case you are wondering, the reason @tkn screenshot looks different than the one I posted is that MATE allows to you select from several different system menu options. Right-click the panel, and select Add to Panel. In the resulting popup, included system menu choices include:
I use Compact Menu. tkn's picture shows the Brisk Menu.
You still can do configure Arctica Greeter through dconf-editor and change the background wallpaper there.
You just need to install dconf-editor
