We have the same problem here in Sweden but only on i6S, and you dont have to replace the screen to get the touch id problem you will get it just by removing the home button and all the flex cables it looks like.
I'm having the same issue, Repaired 21 phones with iOS 10 and 6 of these have given this error. 3 of those we removed with a hard reset. The other 3 we couldn't recover from.
Have you tried a new home button extension flex cable (the cable that runs from the top of the metal plate to the bottom)... I work on these every day (own a repair shop), and at least once a week I damage one of these cables without knowing it at first. They tend to be the most fragile on the bottom end, and if not taken off carefully the small metal plate under the bottom connector comes off.
