Fix screen for iPhone 13 mini

Yes. If you take it to a Genius Bar or authorized Apple service repair shop, they will use genuine Apple parts. A third party repair shop won't.

Make an appointment at the Genius Bar to have it repaired.

Genius Bar Reservation and Apple Support Options - Apple

Yes, if you take it to an Apple store or authorized repair center, it will work properly after repair. They also warranty their work so if it doesn't work right, they will fix it.

What was this "other place"? Was it a genuine Apple repair facility?

If not, there is your answer. Apple does not sell their parts so the other repair shop didn't use the right parts and that's why it worked poorly after they repaired it.