Advise on STS purchase
 

I found a 98 STS on CL for fairly cheap and I have always liked the body style. The owner advertised it needs a water pump. So I went and looked at last night, the body and interior are in very very good shape with only a 107k miles. The story I got is it overheats while driving and he had "mechanic" replace the radiator, that did not solve the issue, his so called mechanic says it must be the water pump. The battery was dead, got it started, heated up fairly fast and the gauge stayed in the middle. I did not drive it cause i was more interested in the motor overheating, let it run for 20 minutes and the top radiator hose never got hot, like maybe the pump is not circulating the fluid, but how is that possible, it was dark but i did have a flash light and could see the water pump pulley spinning, so there is no way for the blades on the pump to not be spinning correct? There was a thermostat in the trunk so that was replaced. The owner is completely oblivious to how an engine works or he was playing stupid. He said he got the car at an auction and drove it home but it got really hot so he parked it till the radiator got replaced and he does not want to put more money into it. He really wants to get rid of the car, I could probably offer him 800 bucks and he would take it. I need opinions on if this is more than likely the dreaded head gasket problem?