Problem with Extended Warranty

You will probably find that the thermostat IS NOT covered by your extended warranty. In the fine print it will say that if a "non-covered" part fails, and causes damage to covered parts, the claim will be denied. As another example, an oil filter is not covered and if one fails and causes your engine to seize, they will deny the claim.
What you need to do is have the repair shop TEST the thermostat and verify that it is in working condition and did not contribute to the radiator leak.