Hi guys, I've registered just for this, LOL.
I think this is an Android 11 bug, not specific to Xiaomi. My MI 10 Lite has suffered since the Android 11 update (aside: I'm not enjoying this phone anywhere as much as my last Redmi Note 2, especially since we now have to use Google apps for messaging, calendar, contacts etc - what did they do that for, I don't want a Pixel phone!!)
I got around it and got my old Moga Pro working in HID mode by doing the following. I am not using gesture navigation on the home screen, I use the old style navigation keys/bar (I have completely forgotten what that's called!!). I am also using the stock global 12.1.4 stable ROM that came via update to my stock phone (no root, stock recovery, no MIUI EU ROM on it yet - not sure if I'm keeping it!!).
1. Connect controller via bluetooth (not just pair, actually connect).
2. Head to Settings-> Additional settings->Accessibility and swipe to the Vision tab.
3. Tap Magnification and turn it on. You should get a little accessibility icon appear in the bottom right of the screen.
4. Tap the accessibility icon that just appeared to turn on screen magnification (you can stay on the same accessibility settings page while you do this).
5. Turn the magnification option off again under the vision tab (hopefully you're still on that page!!)
6. Press some buttons on your controller (not sure if this is essential).
7. Fire up a game and the controller should work again. I've tested my Moga Pro in Shadowgun and the Diii4a Doom 3 pc port and all is good again : ) I haven't rebooted yet so I've no idea if this is a one off thing.
8. Eat toast, frag stuff.
Credit goes to rustybathtub over on the OnePlus7 forum who used a slightly different method and I experimented based on that.
Good luck!!
EDIT: This workaround interferes with one handed mode when you activate it by sliding left or right from the home button. In order to get it working again you have to toggle one handed mode off and on again in settings. Unfortunately, this also stops the game controller workaround, um, working. In order to get THAT to work again, just turn one handed mode back off and run the game. Lol.