A typical flat won't require a new tire, just a new tube. Were your old tires in such bad shape that you needed new ones? If not, then I'd recommend avoiding that shop.
FWIW, debris like glass and sharp pebbles can cut through the rubber on a bike tire. It's not an issue as long as it does not puncture the casing underneath as that is when you get a flat. If you find debris inside of a cut, dig it out so that it does not eventually work it's way through and cause a flat. If nothing is there, just let it go.