If it overheats very quickly, and if it overheats while you are moving through the air, then it's not your fan circuit.
This really leaves two possbilities. One would be a stuck thermostat, which can cause a pretty fast overheat, and the other would be a blown head gasket.
There should be no mystery about this. Both a stuck thermostat and a bad head gasket are easily testable, so there needn't be any guesswork.
If you can get the thermostat out yourself, you can test it by placing it in a pan of hot water and watch it open. If it doesn't open, well there you go.
For the head gasket, the best test is to pressurize the cooling system with a pressure pump, leave it pressurized and then extract the spark plugs and look for coolant on the tips of the plugs, or use a bore-scope to see coolant in the cylinders.
Sometimes the head gasket will not fail in this manner, but instead either a bad gasket or a cracked head will force combustion gases into the coolant.
You can buy a chemical that tests for combustion gases in your coolant.
your mechanic should have been able to demonstrate to you scientifically how he knows the head gasket is bad.