I have a slightly different take. While I won't disagree with anything stated so far, one part of your original post has yet to be discussed. After fifteen minutes of mowing it will bog down and die. That's not a head gasket related problem. Cracked heads, damaged gaskets, and such can cause strange things in the coolant and radiator. Problems with radiator, fan, belt, or water pump can translate to repeated overheating, yes. The "bog down and die" thing is another matter. Is your mobile mechanic aware of that development? If so, and he's looking to replace the head gasket as a solution I think he's only part way there.
I can't see a bad head gasket causing an engine to bog down. I don't do engine work any more, (never been into a Kukje) but when I did that symptom was always a friction related problem. Stopping and cooling it down for XXX length of time before it will turn over fast enough to start sounds like serious problems to me. I've seen lubricated bearing failures do that, but it's usually been piston to cylinder wall clearance problems. When engines overheat pistons expand more than they should and can/will begin to seize to the cylinder walls. If that gets far enough along metal will begin to transfer from the piston skirt onto the cylinder. Once that happens things go downhill fast. I think that's more likely what's "bogging" it down after fifteen minutes. Maybe I'm wrong. Let's see what you find when the head and head gasket are on the table.