A water pump is vital to a car engine’s operation because it ensures the coolant keeps moving through the engine block, cylinder head, hoses, and radiator, and maintains an optimum operating temperature. It is usually driven by a belt from the crankshaft pulley or sprocket.
The answer to the question of whether a water pump can be bad and not leak is yes.
Some modern water pumps have plastic impellers (fan blades) and these sometimes break. This leads to vibrations which in turn cause the pump to fail. It’s also possible for the impeller to become loose on the shaft, so although the impeller looks OK, it doesn’t pump anything!
If your water pump isn’t working, it may be because it isn’t receiving power, so it will not leak at all.