So the service techs started to work on it. The assistant service manager informed me that both tires had puncture marks. I noticed he didn't say anything about the tire protection plan, so I asked about it. He started to say "I don't think it will help..." Then I handed him the invoice/receipt and brochure for the plan. (Good thing I had the foresight to bring all the documentation with me.) It turned out that although I was signed up for the plan, it wasn't recorded in their computer system. After putting it into the system and making a few calls, the assistant service manager confirmed that it did indeed cover the damage. So the upshot is I got two new tires free of charge! (Still paid for labor, but I would expect to pay for that anyway.)
Also on the plus side, their carwash was working this time around so they did the complimentary carwash they owed me from last time.