agreed----- it should pump water immediately.-----Water must be 6" above the pump in a test bucket.------There is no --" stop me / shutdown feature " on this motor.-----You risk damage by not installing an impeller now.----And check the thermostat too.-----I find many of them failed in the open position.
Yes that is the way it is supposed to be.------I would say the thermostat is stuck open or missing altogether.-------Look for it on top of the cylinder head.
How do you check temperature? Warm up depends some on the temp of incoming water. If in a barrel, the water from the hose is about 40 degrees F, while motor when warm should be about 130F, give or take.
It would be a good idea to check t.stat assembly and installation, and test the t.stat itself. The flange of the stat goes into the slotted seal. Function can be tested before installation by dipping the unit in a cup of hot water, or into a pan on the stove, then running under tap water.