Firstly, that table doesn’t show the number of essentially different positions. For example, at distance 2, you have (54, by my count) positions where two faces were turned that do not share an edge and positions where they do.

Secondly, I don’t see why would want to pick a ‘random’ position that way. If those numbers were correct, it would mean a 1:21 probability that your ‘random’ position was zero moves from the solution.

That’s a different problem. If you want to check every possibly you don’t want a randomized cube. You want the full list.

I don't think generating every possibility of a cube is reasonably doable - it's over 43 quintillion - 10^16

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.

For scale, fastest supercomputer is over 1.8 x 10^17 operations a second. So generating 10^16 numbers is not that bad for a distributed project over a few weeks.

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.