You could do that and it will give you the most interesting games.
But the rating system would have to reflect that and take the starting player into account. My guess is that the advantage is 3-4 levels. If you don't account for that, it will mess up the ratings.
For example, if player A is a bit better than B and higher ranked, B will start and win the majority of games as the first player advantage overturns the skill gap. That is inconsistent with B being ranked below A and B will one day have the better rating. Then, A goes first and wins a lot, being the truely better player as well. This is also inconsistent with B being higher rated and B's rating falls again. So you'd end up with swingy ratings that don't converge to any difference reflecting the underlying skill gap.
As you say, the ranking system should incorporate who goes first if each player doesn't have an even chance to go first.
Rather than making the higher rated player always go last, another option is that a player's chance to go last equals the predicted probability that they'll win. This way higher ranking players still get to go first sometimes, but less often the higher rated they are. That method could also be used if the players haven't played before, and if they have played, whoever won most recently goes last. This tends to produce more interesting games, but even a player who almost always plays lower ranked opponents will go first a decent amount.