Though it could be an off chance but it is possible because it has happened with me on a 5s. I was to replaced the charging port of a 5s but in the process of disconnecting the battery, I accidentally broke off a few very small components that I believe were called the F11 capacitor that would be responsible for the charge and how the phone handle its battery life. The phone would die with a charge and being on the charger and would get hot. Look around the battery connection and see if you see any small, and I mean small, round objects just rolling around or been broken off.
Sounds to be the screen. Actually that has happened to me recently with a 6 Plus. The new screen worked but the old home button wouldn't. However it would work with the old screen. I had a replacement sent and should be trying it soon. iFixit parts are good but no one is perfect. I know with the last two vendors that I was using was okay but got worse over time. With iFixit, I have had maybe two times that I have had problems with the parts. But it does sound to be a screen issue. What you could do is move the back cover plate that has the home button cable on it from the old screen to the new screen, if you aren't able to get a replacement. That should allow the home button to work on the new screen.
If you haven't, try a newer replacement part, possibly ones from iFixit. From my experience, when a phone acts weird during a repair with a new part, it could be the new part being defective and the old part being bad. I have had to order newer parts and everything work out just fine. Those sensors are touchy and could be damaged in the slightest possible way. But trying a newer screen could eliminate the idea that the screen is the problem. If that doesn't fix it, there might be a deeper problem occurring.
If it doesn't work after doing a reset or restore, this page links to the repair program of the iSight camera problem of the 6 Plus. On there, you can check to see if your phone fits the ones that Apple shows to have that problem. https://www.apple.com/support/iphone6plu...-isightcamera/
Be sure you have all the updates installed for the system and do a factory reset just to check that it isn't a software issue. If it isn't software related, possibly the parts that deals with the cellular side has gone bad. The wires is a good idea to see if they haven't gone but they aren't usually the common reason. Still worth a shot.
Even if the screen doesn't seem to be damaged, it could be. I had that happen on a Note 4 of a friend. She dropped it many times until one day, she went to use it and half of the screen would no longer recognize touch but the other half would. The stylus would work. But there was no visible damage to be seem. A new screen was installed and that fixed the problem. I believe the screen could be damaged in this situation even though it doesn't seem like it. It is worth a shot to getting a screen and seeing if that fixes it.
It might be possible that a capacitor or filter was damaged when discounting the cables for the digitizer. Sometimes what happens is if the battery wasn't disconnected when parts are being disconnected, parts could be damaged especially when the device is accidently turned back on. Check around the areas around the cables that you disconnected. If anything there would be very small shiny components that could be round on the board. If you see any lose, that will be the problem. But I'm not a board guy so I'm not completely sure on this but from previous experiences, that's what it sounds like.
Try a different charging cable and ac block first just to be sure that isn't a problem. Then try replacing the battery if it seems to be dead. If that doesn't help, you might be looking at more of a component problem internally.
Have you replaced the plugs on it? I'm not sure if that model has it but if it has the spark plug wires, those need to be replaced when the plugs are replaced. If the plugs haven't been replaced, use NGK or ACDelco branded plugs and if possible, pre gapped. It could be misfiring because of plugs, even if they are new. If that doesn't help, it is possible then the distributor is out. But I would check that out first since plugs and wires would be the cheapest, especially if they haven't been replaced.