Chosen Solution
Replacements are rarely close in performance to the original. I have this problem with a first generation IPad Mini (one section of the replacement screen is completely unresponsive). I recommend inspecting the connection between the screen and the phone (if possible). Try checking if the connection is lose or damaged. That in turn, would provide reasoning to why the phone is having difficulties detecting the replacement.
Have you checked to make sure you have not loosened any other connections? You could try trying out a new home button but ultimately it could be a faulty screen or worse… a faulty chip within the phone. Have you tried looking into changing the settings? Could it be that the Taptic Engine is loosening connections? If anything try lowering the response of the Taptic Engine as you continue troubleshooting.
You will need programming pairing which only Apple does, and because you replaced the screen yourself they may not do it because it is Apple