home button sometimes works

There is a very fragile ribbon cable that attaches to the home button from the body of the phone. This cable may have been damaged during the screen replacement that you had performed, and the way to fix this is to open the phone and check and see if the home button cable is plugged in all the way. If it is, then the cable itself may be damaged and you may need a new home button. Note that replacement home buttons do not work with touch ID since it has to be the original home button for it to work. Hopefully this helps.