Fossil fuel engines are significantly less efficient than electric ones... so need a lot more cooling. I'm sure getting rid of the need for that would be a nice win that might be practical with electric engines.

Yes, that's the absolute norm (you often also want to cool the exhaust pipe itself and there you often see coolant water pumped straight into the exhaust). That's the obvious thing to do: You do have a huge reservoir of readily available coolant, tapping it is the sensible thing to do :) That's why I fully expect them to run their coolant through some heat exchange with sea water.