Honda Blackbird sounds good. You do the most obvious, cheapest and easiest solutions first. What he suggests is cheap, easy and obvious, at least according to him. So, I would try what he says first.
In the old days, you can see if the water pump is working all right by opening the radiator cap and run the engine to see if the coolant in the radiator drops when you rev up the engine. If it drops quite a bit, the water pump is O.K., but if it’s bad, it’s not working. Also, if you have cooling problem at low vehicle speed but not at high speed, it’s likely to be a fan problem. This is because at low speed, the cooling system relies on the fan to draw heat away from the radiator, but at high speed, the wind coming through the grill would be enough to cool the radiator. To see if the fan (or components related to it), let the engine run until the temperature rises above normal and see if the fan kicks in. If it doesn’t kick in, it’s the fan.