It sounds like a win-win situation, but unfortunately it looks like at least for Google Stadia, the iOS 15 update seems to have broken it. According to a post on Reddit, a Stadia Community Manager shared that iOS 15 could have made some changes that makes Stadia unplayable on iOS devices, and also how to potentially fix the problem.
Basically, users will need to make some setting changes to Safari by going to Settings > Safari > Advanced > Experimental Features and disable “WebRTC Platform UDP Sockets”. It is unclear if this will be a permanent fix or if it is only a temporary one, but it’s looking like the latter as some users are reporting they’re still having issues streaming games. Some are also reporting that they are seeing gray borders while playing games.
It is unclear what exactly the problem might be and if it is a bug within Safari or if Stadia needs to be updated to ensure compatibility with iOS 15, but for now if you do run into issues, then this is a potential fix you can look into for now.