Thermostat or Not, The water pump always operates at full capacity.
When the engine is cold, The water bypasses the cylingers and is used entirely to cool the exhaust.
When the engine is hot, a portion of the water flows over the cylinders to cool them.
This water then joins the main flow to cool the exhaust.
In either case, the same amount of water is delivered to the exhaust stream.
A small stream of water exits the "Tell-Tale" hole at the top of the drive leg as an indication that cooling water if being delivered to the exhaust.
The Telltale stream is only there to give you confidence that the engine is being cooled.
With out the Telltale stream, there is nothing to indicate that any water is flowing, as the flow starts under the surface and is ultimately discharged under the surface.
Water must be present in the pump BEFORE the engine is started, and remain continuously available until AFTER the engine is shut down.
If you have operated the engine without water pressure on the muffs for more than 5-10 seconds,
it is a possibility that the rubber impeller in the pump may be badly damaged/destroyed.