Since the elecrtic heat is provided through an element that protrudes into the inside of the tank it should work fine without the coolant running through it, but I'd rely on someone with more expertise than me. My concern being that a leak from the coolant tubes might be a precursor to a total breakdown of the continuity of the tank, including the water portion.

Bill, I had that problem. The antifreeze would leak and crystalize near the macerator. I decided to replace the water heater and the heating hoses. In doing this I noticed that the leak was from the antifreeze hoses that was connected to the water heater, all that was needed was to tighten the clamps. These connections are visible only if you remove the cabinet that is below the sink. Check that first.
Visit my website on my signature for water heater replacement and view the album. It is not an easy job.
My favorite Anchorage is Richardson Bay. If you see me stop by, say hi.