Although white has a slight advantage, it is only 'slight.' which consists of an extra tempo which isn't enough for a win. Your idea will probably be enough to give black a major advantage. In an equal endgame, which is very likely to happen in any game. Black can just drop his pawn down on the sixth rank and win due to it being a passed pawn.
Above is a drawn endgame with black (or white) to move. However if it was black to move with your rule in place. black could place a pawn-h6 and then you get the below position.
and black wins because white has to divert his king to the h pawn to stop it and then the black king will have enough time to eat the queen-side pawns.
I was thinking of a way to balance chess more in favor of black, evening out the odds of the first move advantage for white. I was thinking about it and i thought of giving black an extra pawn that she can drop on the 7th or 6th ranks, also black would only be able to do this after the tenth move as this is usually considered the end of the opening part of the game. This could be especially useful in the endgame to block enemy pawns from promoting or protecting the king.