I am not a great mechanic, and I have been fooled by a small bypass hose that goes from the water pump to the thermostat housing. When it leaks, the water drips down just about in the same place as it would if the circulating pump had a bad shaft seal. It is plenty annoying to remove the pump and find that it is OK, but it always fixes the problem because one always replaces the attached hose in the process of replacing the pump.
Now, I do hoses first, and if that doesn't help, then I take out the pump.
1975 32' Flybridge Sedan, twin Perkins 6-354 diesels, 1:1.53 velvetdrives, 16 X 19 props. Merritt Island, Florida