Shop tools required: functioning pressue tester, combustion leak detecter, engine pressure tester, and digital pyrometer.
A cooling system needs to meet four conditions to cool an engine: system integrity( no leaks), Circulation( unimpaired flow, a good water pump), the presence of the correct type and ratio of coolant, and heat transfer( cooling fans operating normally, free flowing radiator).
I have witnessed water pump impeller disintegration to the point where there was nothing left of the impeller but the disk, so yes an engine can overheat due to water pump failure( no circulation, induced by a massive leak( seal failure), a gradual leak, or no leak( impeller failure)
The cooling system diagnostic routine consists of veifying the presence of coolant, presence of the water pump drive belt, topping up, finally pressure testing the system to test for leaks. Leaks present? Determine source with system pressurized, relieve pressure, repair, and retest.
No visible leaks? Perform a circulation test, by disconnecting an engine discharge hose and with engine running, verify loss volume. Volume good? Combustion leak test the engine to assure mechanical integrity. Engine maintains pressure? Replace water pump, thermostat and gasket, flush if necessary, fill with correct coolant type mixture and bring engine to operating temperture verified by cooling an cycling.
Following these steps will result in correct diagnosis, correct repair, vehicle reliability, and customer satisfaction.