Join Date: May 2004
Location: Wilmington, DE
Bikes: 2016 Hong Fu FM-079-F, 1984 Trek 660, 2003 Specialized Hardrock, 2004 LOOK KG386i (RIP), 2005 Iron Horse Warrior Expert, 2009 Pedal Force CX1, 2016 Islabikes Beinn 20 (son's)
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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.