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i think it could at the very least be stated that, in general black and
white play differently from one another.

one way of alleviating this 'problem' - without the use of simultaneous
movement, or in always allowing black 'one last move' - in a given game,
would be to have a chess variant such that the two sides are for all
intents and purposes identically placed at setup, yet the 'black' side
has one piece 'pre-moved' in some kind of logical, standard manner. this
might make up for any real or imagined imbalance.

for example, if you applied this idea to FIDE chess, you could have black
begin with 'king's pawn at 4' (rather than 2) as part and parcel of the
opening setup, yet white would move first.