You're assuming that all the contamination is on the surface, but there is hair, lint, and other particles under your fridge that will stick into the ice.

Also, your example of motor oil was a poor choice. Oil is probably not going to stick to your ice, unless it is in contact long enough to freeze, or get viscous enough to cling to it, and oil will not rinse away cleanly.

Also, how bad to you need to save one ice cube?