I've never had an issue with the quality of the screen being the issue of a home button not working. I have however seen the issue to be a ripped or damaged home button flex or the back plate not being the original which has caused the home button not to work properly.
I've also read that any damage done to the home button will cause the button to never work again as well as even a replacement because it's not the original. You probably would need a new backplate and a new button for the button to be recognizable.
Hope this helps :)