As stated above, I like having that option and bascially agree with what everyone is saying. But aside from the impact on scoring runs, understand that there are other impacts as well. In the end, there are only so many at bats available in timed games or full length games for that matter. More players batting means everyone batting less. If you take turns sitting girls, they will eventually get as many at bats as they would have if you had batted everyone. A girl batting in the bottom of a 12 person batting order will frequently only get one at bat in a timed game ... whereas if she was batting in a 9 person lineup every other game, she will probably get 2 at bats the games she plays. Additionally, it takes away the team's ability to substitute for situations ... pinch runners, etc. And if one team does it and another doesn't in bracket play, the team that doesn't do it and turns that top of the batting order over more often is more likely to move on everything else being equal, and the players on those teams are going to get to play more games and get more at bats.
I'd be interested to hear from SoCal Dad whether they allow everyone to bat out there in tourneys, and whether teams use this or not.