Well, I can tell you the reason there is no family time in the gym for a good reason...
I am a well trained lifter (adult female) and I can't tell you how many times I have seen adults do exercises incorrectly...moves that could cause great bodily injury....
That being said, it gets much worse when adults let their children into the gym to workout...at the gym I belong to, sometimes children try to accompany parents when trainers are not looking and let me tell you I am appalled by what I see going on... Do you know how quickly a child/young teen can injure themselves in a gym?
I don't think this adult only policy is for Disney only. I am sure this has everything to do with the insurance policies in place for the gym....most gyms will not let youngsters participate...for obvious reasons...just because Disney is a family ship does not mean every bit of it has to be for family...that is why there is an adult pool, Palo's, etc. Different activities for families, but not open to everyone....
Before I get a lot of flack for what I have said, let me say I do understand that there are some teen athletes that probably know more about good form in lifting than most adults, but rules have to be set for the majority...Perhaps if your teen needs to seriously train while on vacation, you could speak to someone at Disney and maybe sign a waiver or something to allow your daughter to work out...
But again, you did say you were worried about her burning out on the ship, so again, I suggest some very active excursions to keep her active instead...