iPhone 6's home button has nothing to do with the screen. It's completely separate, the cable too I believe.
What exactly did you replace? The button? Or the daughter-board beneath the button? https://d3nevzfk7ii3be.cloudfront.net/igi/xBsAlmtAuS23eybD.medium
If the button still works, but only "10%" of the time, sounds like it's simply dirty where the connection bridges.
If you can get to the mechanical switch (the clicky thing where the home button presses) when you take your phone apart, put a couple drops of isopropyl on to it (no less than 90%, 100% is best) and click it rapidly around 50 times.
Add another drop, click again. Then reassemble the phone.