I agree the repair shop probably doesn't want to deal with the warranty, as most of these warranties pay fixed, pre-agreed-upon repair rates. They would rather charge you market rates. And of course, the warranty company would rather not pay anything.
So, if you get the repair shop to agree with the warranty shop that it was a stuck thermostat that caused the problem, you are stuck.
I would ask for the original thermostat back. Do not ask why, and don't wait or it will get "thrown out". When you get it back, test it with the boiling water test above. If the thermostat opens and closes in boiling/cold water, then you prove that the thermostat was functioning properly. You can make a video of this test and post it to YouTube and share it with the relevant parties (and the rest of the world).
Everything rests on getting your hands on that thermostat before they have a chance to "lose" it or monkey with it.