Are you sure? I don’t think so. The thermostat runs to keep the engine from overheating, so running the heat brings the air away from the engine and under the hood.
According to freeengineinfo.com, Thermostat replacement is generally done if a car is overheating. The other possibility is that the car is no longer heating up correctly due to the thermostat sticking.
Meaning if the thermostat isn’t working (or isn’t there) the engine will overheat, not run cooler.
If the thermostat wasn't there, the coolant would flow all the time. When most thermostats fail they open up so the engine is protected (although your heater won't work and your mileage will stink). Occasionally they freeze closed in which case you need to stop the car before it overheats. All the ones I've had to replace have failed in the open position which required a very cold drive to the parts store (why do they always fail in the winter?).
A thermostat does not “run”. It is a fixed heat sensitive valve.
Thermostat doesn’t regulate air. It regulates water.
A thermostat is a spring loaded valve that is in-line from the radiator to the engine. When heat builds up the thermostat begins to open allowing cool water to flow to the engine. Engines (especially fuel injected) require a precise engine temperature to run efficiently. The thermostat regulates the amount of water that flows to the engine.
If a car is over-heating, yes, it could be the thermostat. But that means the thermostat is stuck closed and will not allow cool water to reach the engine.
The thermostat opens when the engine gets up to temperature.
A stuck thermostat remains closed, preventing the coolant from flowing to the radiator, where the airflow cools it.
No thermostat means that the engine never gets to operating temperature, and because the coolant is flowing so fast past everything, it can actually cause hot spots to develop and the engine to fail.
You got lucky.