Further to my Xbox One X observations, I thought you might like to know that I tested on Quest 2 today using the Quest's inbuilt browser. I only tested Donkey Kong dual screen, and it ran ok but did not accept any input from my bluetooth connected Xbox controller even when I attempted to remap the controller buttons in the game settings. It may work with a bluetooth keyboard connected, but I dont have one for testing unfortunately.
I might try and sideload another browser to the Quest 2 such as the Chrome apk version and see if that works any better with a controller.
Edit: I managed to sideload Chrome apk, but results are even worse with regards to controller, still no ability to remap Xbox controller buttons after connecting it with bluetooth, but sadly also lost the ability to press the buttons on the game itself with my Quest controllers (which I can do with inbuilt Quest Browser), I can press the play and other menu buttons with Quest controller, and grab and rotate the device but can't press the onscreen buttons on the game itself. Oh well... hope this helps someone else attempting to do the same.