I took the heater assembly apart, drew up a schematic, and see that the heater takes the full 110 V AC, and is switched on and off with a relay controlled by the PC Board. The heater element itself measures about 15 ohms, which seems reasonable. And the transformer steps down the 110 Volts AC to 12 Volts AC, which is then rectified to provide power to the PC Board. Pretty simple circuit.
Don't know what 'Poser the element up' means, but assume you mean wire it straight up to 110 Volts AC. Nice to know that may burn out the element instantly. Thank you for that. I won't hook it up straight to 110.
So now I'm thinking the heater element itself is ok, since it looks clean and the resistance is at a reasonable value. So that means the control circuit is bad - could be the PC Board, the temperature sensor, a bad relay, who knows? And I don't have the tools to troubleshoot and find out what is bad.
And I'm not going to contact the manufacturer/dealer, since they are NO HELP, so may go to a pool place and see if they can build me a custom system or something. That will probably cost a lot, so I am probably stuck with a fancy inflatable little swimming pool, unless I can figure something else out. Any ideas are most welcome, and I appreciate the advice.