Came back to mention that it worked well for a couple hours after disabling Picture in Picture mode, but it is back to lagging again.

Hi everyone, I know this is an old thread but I was still having this issue on iOS 15 on an iPhone 12 Pro Max. Someone mentioned to turn off the picture in picture setting and that worked for me. No more lag in games and the interface. Hopefully this works for others as well. Will report back if the issue returns.