You are right, the hot water valve is stuck but it is inside the climate control valve. I had 2 of them go bad in 2 different cars. One I replaced and heat works automatically to a set tempereture and the other one I replumbed it and put in a hand operated valve but it is difficult to adjust the temperature.
As far as the aux waterpump is concerned, it is only needed at idle to get enough heat in cold weather. System will work at road speed even if aux pump isn't working. Usually the electric motor in the aux pump burns out and the impeller does not rotate but the coolant still passes thru it.