I don't know the first thing about dishwashers, but it seems obvious to me.

The cycle-to-volume ratio is as bad as it could possibly be. Conventional dishwashers recirculate water as they wash and rinse. I imagine there's an mx + c formula to how much water is needed (c = enough water to prime the pump or whatever). So compared to a normal size load, you'd be wasting that constant amount of water.

The wash is also likely going to follow mx+c (c = time for grease to break down, time to rinse, time to dry etc). You can wait a few hours for a whole set of crockery. Can you wait a few hours for a single plate?

Commercial "passthrough" dishwashers work very differently. Manual mechanical action with a spray, plus a quick wash, sterilise and rinse. At that point why not wash your single plate by hand?