Please forgive my ignorance, remote GUI login is something I know very little about. So in the quote above, would this be a minimal Mate desktop set to boot to the command prompt, or a Ubuntu server with a minimal Mate desktop installed?
Ironically, I recently implemented multi-user target on a Mate desktop that I had installed on top of Ubuntu 16.04 mini. It was purely experimental, but one thing I noticed by using multi-user target was that it really wasn't like the machine booted to the command prompt and stopped there. To me this was more like booting into a GUI text mode, than it was booting to a command prompt. And during shutdown I also got the mate graphical splash as you wrote in your original post. This is probably part of the problem your having, as you have already indicated.
I also looked at the NoMachine web site. I may have to experiment with this when time permits. Anyway, if you had the minimal Mate desktop set according to plan, would you log in to it using something like this?
Edit:
Not being satisfied with the Mate desktop mentioned above, I eventually started over and found a way, with a tip from @lah7, to install an extremely light version of Mate on top of Ubuntu mini. It might be of interest to you as there is no need to use multi-user target, and it boots to a true command prompt, using startx to run the GUI. Then back to the command prompt when exiting the GUI. Would be good for a lite version of Mate on a server if someone wanted to do so.
https://ubuntu-mate.community/t/lightweight-desktop-environment-for-ubuntu-server/13479/11?u=steven