water pumps rarely fail to pump
true, nut it needs to be checked if you have cooling problems. As the engine is now over 30 years old, you don't really know whats happened to it. If the car has been run without antifreeze 2 things can happen:
1) severe corrosion (the pump blades get eaten away)
2) freezing - (the remains of the blades get stripped)
I've seen a couple of cars (A-series and a Ford) where the pump had no blades left and both times the car had been run without antifreeze.
It is extremely rare, but the older a car gets and the less well maintained it is, the probability increases.
You can check by seeing how high the pump can push water up the heater hose but without 2 cars next to each other it's not easy to do an A/B comaprison (even a pump with no blades will push water into the rad)
But if you have the time and inclination, just take the pump off to inspect it - you'll need a new gasket but it's better than buying a new pump on the off-chance.