you are correct on the working of the thermostat valve, it blocks coolant flow to the radiator until its set temperature is reached.
The radiator fan is controlled by a thermoswitch, and it will engage the fan anytime it comes to it switching temperature usually about 92-96 degrees celcius. the operating temperature will be reached eventually - even without the valve, but would take 3 or 4 times longer, which is terrible for your engine. So its in the best of all interests to put a working valve back into service.
the reason you may be seeing water getting sucked down before must be because of the partially blocked return flow of the cooling system, as you have zero blockage now you are seeing vigorous movement of water.