Hi Mark,
Assuming this is a TC, the last point about "stayed cool on the last leg" might be because the water level had dropped below the temperature sender in the head. You're measuring air temperature at that point so all bets are off. The missing thermostat is a concern, no reason not to have one there and maybe it was removed to mask a high running temperature in the car's previous life ?
First check is as BDA says, look for rust stains at the water pump weep hole, a sign that the bearing or seals are failing. If the seal goes then it's a serious leak and you can't miss that one. If the engine has been standing for a long while then there's a good chance of rust in the system and maybe that's worked it's way into the pump. But I'd have thought you'd need a lot to do any real damage so that would be low on my list.
The belt doesn't need to be that tight to turn the pump but when you say you can turn it by hand, can you feel any play (up/down/sideways) in the bearing ? It should be rock solid, no play at all.
If that's ok, get the car running and rev the engine while feeling the top hose, that should show an increase in temperature and give an idea if the pump is pushing water into the transfer pipes. If that looks ok then the same check on the inlet/outlet hoses on the radiator ? A slower rate of increase but should be measurable by touch.
After that I'd be looking for compressions and signs of head gasket failure, coolant system pressure, oil/water/etc. That could account for throwing water out of the overflow but not the cool radiator, unless of course it's happening so quickly that there's no time for circulation.
Going back to the belt, because they are hidden away we tend not to check them as often because, well, you can't easily get to them. But they can wear and snap (mine did once) or maybe yours is glazed and slipping when revved ?
Brian
