US dishwashers often do have a wimpy heating element. It gets used if you have a setting for extra hot water or a sani-rinse cycle. It wouldn't be a big deal to have the dish washer heat it's own water, but it would require either running a 220V circuit to the machine to make it practical or put up with a very long wash cycle.

Though frankly, washing in cold water is fine as long as you scrub well enough. But you do need to do the rinsing in very hot water to finish the job.

Edit to add - I didn't explain the extra heating to the water well enough. The wimpy heating element doubles to do extra heating of the water and is used for the drying cycle.

It doesn't have to, just put a heatig element in it. Most dishwasher I've seen only hook up to cold water, so running the tap is not going to do anything.