Phone won't turn on

I'm going to do my best to diagnose this step by step.

Your phone stopped working all of the sudden, without water damage. That's usually the sign of a failure of some part of the motherboard.

iTunes didn't recognize it but your PC did. That is another sign of motherboard problem.

After a flash, iTunes does see it. OK, that's good. That means that the iPhone is somewhat working.

Phone doesn't make or receive calls, screen doesn't work.

That's going to be the symptom you focus on. My first instinct is to try a new screen. I can only think of 2 things that would cause your issue: a drop that damaged something or knocked something loose or a corrupt memory. The latter isn't really common, so I'd lean towards the former. So if its drop damage, that's pretty easy to inspect. The first thing is to try a new screen. If that works, great. If not, then you will need to inspect the board for loose or damaged ICs. iFixit has a great youtube video showing how to do that.

To sum it all up: It sounds like you either have a bad display or some board level issues. The bad display would be the easiest fix, so try that first and then update us.