Is it also not true for puzzles with equivalent pieces? Since not all permutations are in the scrambler
This is a bit tricky to answer. The key observation to make is that, even on puzzles with equivalent pieces, every possible move is a permutation of all the possible states—in other words, that every move can be undone. After applying a random-state scramble, the probability of being in any state is equal (= 1 / (# of states)).
Applying any fixed move after a random-state scramble only shuffles these probabilities around, which effectively does nothing at all since they were equal to begin with. Likewise, applying any fixed move sequence also effectively does nothing. Consequently, applying a move sequence chosen at random from any set will also do nothing.
On the other hand, if you apply random moves
before the random-state scramble, then on puzzles with equivalent pieces, this is not necessarily the same as just doing one random-state scramble. The above argument doesn't work because you don't start in the stationary distribution, and you can't just "take the inverse" to force the above argument, because puzzles with equivalent pieces (usually**) don't have a group structure.
** I think there was some puzzle that was isomorphic to one of the sporadic simple groups despite having equivalent centres. Can't remember what it's called, though. (Edit:
it's the trapentrix, which isn't one of the sporadic simple groups.)