That’s a different problem. If you want to check every possibly you don’t want a randomized cube. You want the full list.
Edit: I think you simply didn't understand what I was talking about. The distribution of states is not even. If you randomly perform moves on a cube then you're more likely to end up in certain states.
As to the distribution of states, I am not sure what you mean. Insufficient shuffling can introduce bias, but that gets reduced as you continue shuffling. You can trivially shuffle past the point where remaining bias is not detectable. Unless, you want a specific bias, then that’s more easily achieved generating a random number with a specific format.