It's not the tire rubbing the brake pad but the rim (outer circular metal part of the wheel). Truing a wheel (fixing a small area or areas that hit the pad) is a separate job from flat repair, but the shop should have pointed out the out of true condition, and offered to fix it either at the time or by you leaving the bike. If the rim rubs the pad all or most of the way around it's not truing that is the problem but incorrect installation of the wheel, which is definitely the shop's error.

It's always a good idea to look over the bike and if possible ride it before leaving the vicinity of the shop. If you don't want to fix your own flats that is fine, but I highly recommend that you have someone show you how to do a routine "quick check" of your bike. You can also Google "bicycle quick check" - there's at least on video available.