While Steam Input is turned on if my controller comes unplugged most games reports a "Controller Unplugged" with the prompt to press any button to plug it back in. When I reconnect the controller it can operate the Steam overlay, but the game doesn't register input.
Further details:
I'm using a mix of controllers (Xbox One, PS4/5, Switch) all using a magnetic USB cable, this occasionally comes loose causing the disconnect.
This is issue in Cyber Punk 2077, Binding of Isaac, Deep Rock Galactic Survivors, Power Wash Simulator, basically any game that has Steam Input enabled.
If I disable Steam Input for that game the controller reconnects just fine, but in some games this makes the game unplayable as they lean on Steam Input.