Can’t you fix the tires?

This morning, after 35km, I noticed a semi-plated tyre on my A3 (photo). Driven directly to the petrol station and filled it up. Searched the tyre and found a screw. Then to the tyre dealer. He says that it is not repairable, because the spot is in the shoulder area (he said nothing about the speed index Y). Is the screw really so far outside that the tyre cannot be repaired? At the dealer I am not quite sure if it can be repaired. With the prices. 113€ he wanted for it (incl. assembly). 65€ costs ́s on the Internet (Next N8000). And for the assembly alone 24€ The tires are by the way only about 6 months old (approx. 7000-9000 km), I think there should be a new one in case of a new purchase. And this then on the non-drive axle.