Plumbing Questions? Ask a Plumber and Get Answers ASAP.
It is a bit surprising that in all those visits no one has figured it out.
The odds are in the 90% range that hard water scale has built up in the heat exchanger and restricted the hot water flow, so at at the shower for instance, cold water pressure which is then higher than cold water pressure over comes the hot water flow, stops hot water flow and then the flow switch in the water heater shuts the burner off.
We can do a few tests.
Turn the shower on wide open and to hottest water, next turn on the kitchen sink hot tap in the same way, and then tell me if the water heater flame ever shuts off and you get cold water at the taps or not, we can go from there without time limit.