Typically the manifold oring will leak when you turn the vehicle on or off, squirting fluid out under the manifold. If you look where the pipe comes out of the water manifold with a flashlight, I will bet you see a little (or big) squirt of fluid when you stop or start the engine. It is very unlikely that the thermostat is leaking - it will be a first I’ve heard of one leaking in over 5 yrs of these boards.

How long are you running the engine before you feel the hoses? In the cold it will take more time before the thermostat opens. If after 10-20 minutes or a long drive that the heater does not work well or hoses are not hot, then thermostat has failed in the closed position which is rare since they are now designed to fail in the open position. And even if the thermostat fails, that would not cause a leak. But if the gasket is bad, then replacing the thermo at that time is smart. How much is he going to charge to pull the manifold?