Bad/Weak water pump?????

The thermostat sets the floor of the cooling system, not the ceiling which is your problem. I would try another, not remove it. There can be more harm than good with no thermostat to balance flow through the block. Check the radiator fins. Clean them out with compressed air. Simple things 1st.

Nobody said leave it out, but just test without. :D
Doing the thermostat AND water pump with a flush would be the best start, or at the very least pull it for an inspection and still flush it out.
Have you seen some of the pumps with the vanes rotted clear off?
Nasty.