The Backbone One would be an ideal game controller—if the iPhone had more games
Yes, you can sync popular Bluetooth controllers from Sony, Microsoft, Nintendo, and 8bitdo with your iPhone, but that's not really the ideal answer—your iPhone isn't a big TV sitting across the room ...
5 great things you can do with a Google Stadia controller
Many Bluetooth controllers work for mobile platforms, Chromebooks, PCs, and consoles, but a good one can cost $50 to $80.