Perhaps try updating the Xbox controller? There's an app through Microsoft that lets you update the firmware on the controller.

If you add the game to Steam (using the add a non-Steam game to Steam function) and then launch it through Steam, that may work.

Try to put out other USB Devices, my Guess is, that the PC recognizes another Devices as a Controller,
so that you actual Controller is the "second" one and the Game does only take Commands from the first one.

This can be caused when Steam is running or because a Driver makes strange Stuff.

So just let the Mouse and the Controller plugged in and put even the Mouse out after starting the Game.
If it works, put one by one in again and maybe you can spot what is causing it.

Tested just about everything I could. The best I could figure out is a Windows 11 issue?

Changed Steam to not launch at start in both the Steam launcher as well as in my Task Manager.

Tried controllers in other games on GoG, that all worked just fine.

Tried Xbox Elite Series 2, a PwerA knockoff Xbox controll, and a Switch Pro Controller. They worked just fine in Steam and elsewhere just only the Start button was accepted in BoF4 on all three controllers.

I tried classifying BoF4 as a game for Xbox Game Bar to acknowledge and even lauch the game via this method so that I did not use mouse or keyboard to lauch. Still only Start button.

I unplugged all devices except mouse, which I promptly turned off once the game launched, still nothing.

Finally I figured I should try this game on my Windows 10 PC, and voila, it works just fine. Windows 11 computer is less than a year old and doesn't have much going on in terms of other softwares, applications, or background programs. The Windows 10 PC is like 6 years old and has lots of junk on it. Heck I left all peripherals plugged in and let Steam autorun at launch on Win 10 and it still worked there. What gives?

Not sure what to do with this information but there is some issue between Win 11 and 10. Same controllers, same method of launching the game, but just doesn't work on Win 11.

Edit: I tried unpluggina my 2nd monitor, also tried compatibility modes. Neither of those worked for the Windows 11 machine.

Pardon me if this has already been mentioned - but my wireless controller also does not work but I found if I plugged it in and make it wired it does work

Try plugging in your Xbox Elite Seires 2 via USB and see if it works.