I wonder if the taptic feedback you are getting when you unlock the phone is more related to successfully registering TouchID than to recognizing the "click" action. For example, if you could delete the stored fingerprints, the button may not recognize your finger push at all when locked. Clearly the button doesn't register your clicks afterwards.
This scenario happens when the IC (Turtle / U10) on the Home Button flex gets damaged. This is a common occurence (relatively speaking) when the button flex is transferred over to a new screen. The solder balls underneath the IC can crack and cause connection issues. This can be repaired via a micro-soldering repair.