The Control Center's GUI closes upon the player's clicking any toggleable buttons, and it is just sort of annoying. Can it be made to keep the GUI open?

In your opinion... probably something you should discuss on discord or GitHub Discussions to see what other users think as I don't have an issue with this behaviour.

Personally I think closing it is better, because most feedback from button clicks in the control center comes in the form of chat messages, which are pretty hard to see with the GUI open.

Player preference?

Might be a good idea, git cherry-pick is starting to become fun to me at this point. Backporting changes to 1.16.5 is my new addiction.