I've done many 6, 6S, 6S+ screen swaps and never had an issue. I never had an issue with error 53 last year and even documented tearing down multiple phone, using multiple screens and different home button flexes. Its not the screens and its not iOS.
Disconnect the battery before you disconnect anything. Reconnect all cables, ESPECIALLY THE HOME BUTTON CABLE before you reconnect the battery and you won't have issues. I have an iPhone 6 sitting next to me that I've swapped out nearly every single part on, short of the actual home button, and its runnion iOS 10 and Touch ID works fine. Its even got the cheapest screen I could find, just to prove that the screen doesn't matter. Its also got a replacement home button flex that I ordered from China, again, just to prove that isn't the issue.
Saying there is some mass error just because of 2 phones is just silly.