I'll just spitball because I do not have these problems, although I do have quite some experience with Steam thinkering and troubleshooting. Don't take my answers as actual solutions.

Controller inputs break constantly. If I use big picture mode, the games will be in some quasi state where it both detects my controller and doesn’t.

When you are not using an Xbox or Playstation controller, and Steam is configured for it, Steam will emulate it using a secondary spoofed input device. This might be your problem. Compare your input devices with Steam closed and opened. Look under the controller settings and thinker with Steam Input for generic controller.

The right click menu in the system tray does not work 30% of the time.

It always works for me on Gnome. I guess this might be a desktop environment problem and not a Steam one. What DE are you using?

Worst of all, when I close a game, steam will not detect that it closes.

I've had this kind of problem when making a Flatpak Qt app that launches another program. I had to specifically spawn the process with flatpak-spawn to not run into flatpak issues. You say that you also have this problem on a native install, but maybe you didn't really start the native install or you are mistaken? I've mistakenly troubleshooted what I thought to be a native process but I was trying the flatpak I also had installed.

In game, the shift+tab menu does not work; half the buttons are unresponsive and the game will crash upon closing shift+tab

I think this problem is because of the same reason as the previous problem. It might not be spawning processes correctly so it borks the hook. Or maybe you are using some very strict memory protection.