Unable to modify display settings (GUI fails silently)
Solution 1:
I ended up solving this issue myself. The solution, for me, was a program called arandr. arandr, as far as I can tell, is a GUI for xrandr -- the command line tool for managing monitor layouts.
I installed the program with:
sudo apt install arandr
It was easy to use and worked perfectly. I'm still not sure why the built-in tool doesn't work, but I'm happy to have the problem solved.
(Presumably, you could also solve this using xrandr if you know what you're doing. Unfortunately, I don't.)