Chosen Solution
I recently replaced my screen and when I first turn it on everything works fine, but when I turn off the screen and then turn it back on, everywhere I touch just brings up the pull down menu.
To anyone facing this issue, the problem seems to lie in the replacement screens themselves. A workaround is to go into Settings > System > Gestures and turn OFF “Double Tap To Check Phone" After turning this gesture off, the phone works normally.
This sounds like a faulty display assembly or possibly even just a poorly seated cable. If you can, try to pop off the display connector and give both sides a gentle clean with some high % isopropyl alcohol. If the screen is still unresponsive, it’s likely to be a bad display assembly. And unfortunately if the same behavior is present on multiple displays, it’s possible the board was damaged somehow during repair, in which case, you may want to look for a new device or send it to a board repair service.
TO the original poster, I am facing the exact same issue after replacing my broken screen, were you able to find a fix for your issue?