If DSX has an active profile that makes use of Xbox 360 emulation mode running, then the controller is shut down, and then its turned back on (seconds, minutes or hours after, as long as DSX is still running in the background) DSX will create a secondary gamepad entry (as far as I can tell, will explain further below), which results in the PS button not opening the Xbox Game Bar overlay, and certain games not detecting the controller at all.
To check what was going on, I used the DuckStation PlayStation 1 emulator's controller settings since it updates it's controller entries in real time (it makes use of SDL). Once the gamepad is turned on, it will create the following entry:
SDL-0: DualSense Wireless Controller
Which is correct, because HidHide didn't kick in yet. After it does, that entry dissapears and changes to:
SDL-0: Xbox 360 controller
That's how it worked as always, but then that entry dissapears and changes the SDL-0 entry to SDL-1, suggesting a different controller (for a second player) was plugged in. Windows also plays the device connection sound effect when that happens. This results in the PS button not opening the Xbox Game Bar overlay, and some games not detecting the controller at all.
A momentary fix for this issue is switching emulation modes from Xbox to DS4 or DualSense, and then setting it back to Xbox. The PS1 emulator mentioned above will report SDL-0: Xbox again, Game Bar will open as usual and games will recognize the controller as always.
DSX v3, Windows 11 23H2