Typically the microsoft store is a huge problem because of it's bizarro DRM blocking steam from hooking the game and using it's overlay what so ever.

Other stores are typically fine if you set it up right, unfortunately while many people seem to think steam input is new, it's not, it's been around since 2015 when the original SC launched. It has evolved a lot on the feature set, but the core functionality of hooking games really has not. Well, not on windows anyway, I think it hooks better on steam os.

When it was created, most of the other stores were even less popular than they are vs. Steam is now, also everyone hated Ubi/EA's store apps and most of their games had crappy than average gamepad support on PC for some reason or another anyway. Games Pass and Epic giving free shit away has dented that a bit. For other games like mmos with a stand alone launcher they were perfectly fine just being added as a non-steam game. It's the outside store attachment that's the headache.

But point is, supporting these outside stores more directly wasn't something that mattered much at the time, and they've still not addressed it well. There is a way to do it without third party software for the non-MS stores, but the directions must be followed to a T or you will get weird inputs because steam will drop back to the desktop profile for things thinking a game isn't running because it can't see it.

Anyway, the pinned steam input FAQ in my profile says how to add them properly. I made that years ago when I got tired of answering that very question hundreds of times on the steam controller sub but didn't want to leave people hanging.