By placing all those caveats and limits on it, I was trying to get at the PRINCIPLES, as I always do. I'm not interested in emotion.
So, in other words,
1) Are sports completely unacceptable for ladies? You said non-competitive sports were OK.
2) Is any competition between ladies OK? Playing board games? Playing competitive chess against other ladies? Girls competing to get the best grades and become valedictorian at an all girls' school?
So, if sports are allowed, and competition is allowed, what is it that makes competitive sports disallowed?
I understand that excessive physicality in a sport is certainly against feminine nature, so for instance, sports like football and hockey, etc. And it's certainly not feminine for girls to play against boys, since this militates against the future attitude of being deferential and submissive to their husbands. And physical competition against boys adds the additional aspect of inappropriate physical contact between the sexes.
But what makes it wrong to practice and then play against another girls' school. Is it the competition? Is it the sport? ... as per questions #1 and #2 above. I mean, it could be a healthy way for girls to explore competing against one another so that it doesn't manifest itself in sinful ways later on, e.g. competing with other girls to catch the eye of a man, even if it means immodesty. Women DO in fact compete with one another for husbands. There's no question about that. But what if they understand how to win graciously and to lose graciously. Having "practiced" losing and winning, perhaps they would not be so afraid of "losing" at something later.
You learn in sports that it's not a satisfying victory to win unless you play by the rules. So, perhaps a girl could learn that it's nothing to brag about winning if you didn't play by the rules ... and won only due to violating the laws of God.
Another thing I find potentially quite valuable in girls' competitive sports. Girls by nature tend to resent their OWN TEAMMATES if they outplay them or are better. This gives them a venue to come to terms with those emotions, and to learn to be happy for the TEAM if their teammates perform well, a lesson that's only learned with great difficulty by girls.
But enough of the potential benefits. I cannot in principle come to the conclusion that competitive sports between girls is inherently unacceptable and harmful ... if they're not particularly aggressive forms of competition.
To my knowledge, except for potential reasons of MODESTY, the Church has never cautioned against women competing athletically against other girls. And all the caveats by Pius XI regarding female sports has to do with whether they take place "in public". That clearly implies that they're OK so long as they're not carried out in public or the girls are dressed modestly. And I added both of these stipulations in my initial response.