Screen unresponsive after replacement

I think the problem may be that only Apple and Apple authorized repair centers have access to OEM new parts for service. So third party shops are using salvaged parts, counterfeit or poor quality knockoff parts, not new.

Only you can decide if it is worth paying again, although you might want to try a different shop - one that actually does stand by their work and doesn’t shrug and tell you your out of luck when their parts or their work fails.

Apple won’t service your device now, so your only option from them will be to buy a new phone.

It really depends on what's wrong with it. There are certain types of damage where Apple will refused to perform a screen replacement, such as a case that's damaged to the point where they can't properly seal it to factory standards or if there's something damaged internally. A 3rd party shop may not care if it's up to factory standards.

I don't believe that there's a blanket denial of repair service based on going aftermarket, but it's something that's taken on a case by case basis. If it's been run over by a car, then there's a pretty good chance that it's damaged in a way that Apple would have only "repaired" it as a complete service replacement device swapped for the damaged one.

Since you had a 3rd party shop replace the screen (and apparently not a reputable one) it is doubtful that Apple will let you do an out-of-warranty replacement of the iPhone but you could make a Genius Bar appointment and ask. Otherwise it is time for a new one.

BTW, while going to Apple may cost more out of pocket for a screen replacement, Apple will always install an entire new OEM display and they guarantee their parts and their labor for a minimum of 90 days. So often, seeking third party service is not the bargain it seems to be, and Apple’s repair prices are not as bad as they may seem given they stand by their parts and work with a written guarantee.

If you are going to use a third party repair shop, ask them about their written warranty for parts and service before you get them to do anything.