[COLOR="Blue"]Note - Kitchen sink/dishwasher is long way from hot water heater. We always use high temp wash = on. Heated dry = off.[/COLOR]
Your problem may be due to this re: the heater is on for a much longer time than it would be normally as it tries to get the water up to temperature.
May even be that the board had a cold solder joint to the relay. The electricity sees this as a resistance so it heats up at that point which further degrades the connection which causes more heat and so on and so on till the connection fails.
Run the water in the sink which I am assuming is close to the dishwasher.
Check how long it takes to get hot. There is not much you can do if it takes an excessive amount of time. You could also check the temperature in the dishwasher just after a fill (before the heater comes on) to see how many degrees it has to heat the water up to You could try shortening the hot water supply line to the dishwasher but that may not be possible. You could insulate the hot water line to the sink/dishwasher from the tank.
Once the water is hot at the sink check it with a thermometer, it should be a minimum of 120 degrees F.

[COLOR="Blue"]So - looks like I have to replace electronic control. I am considering new Thermal fuse kit 8193762.[/COLOR]
I would replace this.

[COLOR="Blue"]
I am expecting either a shorted heater or more likely a low value such as 4-6 ohms instead of the 10-30 ohms it should read (per schematic).[/COLOR]
I doubt you will see a problem here. The heater is basically a straight piece of resistance wire so cannot short to itself to cause a lower resistance or a short.

I would check the thermistor and make sure that the connections to it are nice and clean. As you can see it's resistance lowers as temperature rises so a dirty connection can cause the heater to run longer than it should.

Might also be a good idea to replace the hi-limit thermostat just to be safe.

[COLOR="Blue"]Duh - nothing bad?[/COLOR]
Yes everything also looks OK to me

[COLOR="Blue"]What is blowing the board?[/COLOR]
I cannot say.
I can give you a possibility.
A bad connection is a resistance to electricity and will heat up when current passes through it. This then degrades the connection causing more heat when there is current and so on and so on till the connection fails.
A long shot possibility is that the relay contacts pitted and was the bad connection. This is a long shot as in mots cases they would weld themselves together.
A more likely scenario is the solder connection on the board. If it was a cold solder joint that could be the cause. Also the relay actually moves a little every time the relay closes/opens so if the solder joint was not good it could come loose over time and you then have a bad connection.

In either of the above it sounds like the heater has to be on longer than is usual due to the long water line so although it should not cause the problem it will make the problem worse because your heater is on longer than what is normal.

[COLOR="Blue"]Why did the TCO NOT OPEN?[/COLOR]
Again I can only guess.
If the heat was localized away from the TCO it would not open.

[COLOR="Blue"]Why was it willing to keep running after a reset?[/COLOR]
If the relay was just causing an error re: the water is not up to temperature then resetting it would clear that error.

By the sound of it you have fixed the problem.
I would give it a week or so and then open it up and see if there are any signs of it overheating again.

If OK, then i would also do this after another month of running.