A sidewall puncture is almost always a death sentence for a tire, and the shop is right. The sidewalls (the sides of the tire) aren't designed to be repaired: there's no reinforcement with cord layers like there is on the tread, so when you're driving, the damaged area will keep working and delaminating. Even if it seems fine right now, the puncture will quickly spread, and the tire will literally burst while you're driving - that's dangerous 🚗
How they determine if it's irreparable is simple: if the puncture is on the sidewall or close to it (usually roughly up to 25-30 mm from the edge of the tread, though it can be stricter), they won't attempt to repair it. A hole in the tread down the middle - okay, you can patch that, because the pressure is distributed evenly there. But the sidewall carries the entire load at an angle, the tension there is enormous, and any patch just won't hold up.
Here's an important thing people often miss: the problem isn't just the puncture hole itself, but the damage to the edges around it, which you mentioned. If the sidewall's integrity is compromised over an area bigger than a pinhead, repair is already impossible. So if you want to make sure they're not lying to you - ask them to show you the puncture exactly, where it is and how serious the damage to the edges is. A responsible shop will show you without any problem.