First, on the phone, you aren't talking to the guy who is doing the work - you're probably talking to a service writer. Further, they do so many cars in a day that what happens to a particular car doesn't stay in their memory.Mechanics and shops don’t want to hear how to repair a car from their customers. We did discuss the possibility of having to replace them if they can’t fix them, but they don’t think it’s necessary at this point, whereas the first shop told me I needed 3 and if I’d have bought from them, I’d be getting 4 tires since 3 failed.
So ride the car before you take the next step. It might actually be fixed.
If it isn't, take it back. If necessary, involve the vehicle manufacturer's District Service rep.
And a bit of clarification. When it comes to vibrations, warranty is a bit complicated. Technically, the vehicle manufacturer doesn't cover tires, but they do cover the vehicle - which includes the tires. What should happen is that the vehicle dealer would do what he can to fix the situation, balancing, match mounting, etc., but if he needed to replace the tires, the dealer should do that under the vehicle manufacturer's warranty. Standard procedure in these cases is for the tires to be sent in for warranty adjustment, just like any other part - and the consumer isn't even aware of this happening.
But some vehicle dealers don't have the capability to do tires, so they claim the vehicle warranty doesn't cover tires - which is sort of true. What the vehicle manufacturer doesn't cover are road hazards, wear, and tire manufacturing defects - BUT - they do cover vibrations and shipping damage (flatspotting would be shipping damage!).