Just before my dd turned five, I moved her from her "fun" gym to a more serious gym. This was a huge mistake and I made it because I thought that she was small, strong, flexible and loved gymnastics. What could go wrong? Everything. The expectations of maturity from a five year old were much too high and ended with ZZ shutting down and with the gym asking me if there was "something wrong with her." I took her out of gymnastics for six months and then took her back to her fun gym where she was happy and once again loved gymnastics and her coaches. My original mistake was in not paying attention to the clear signals my daughter was sending. She was clearly uncomfortable with the new gym and I failed to pay attention.
We moved to a new city in January and my first criteria for a gym for her was that the atmosphere be positive and that ZZ love it. She turned six in November and has matured a lot in the past year. She listens to corrections and doesn't think that the coach is mad at her when they give them. She is learning but still needs a coach who will pick her up and tickle her and let her know that they see her as a person.
I guess the moral of this story is that I've learned to pay attention to how my daughter feels. It may be that your dd will just want to move slowly along with her rec class and private lesson for the time being until she is ready to commit more focus to serious gymnastics. I don't think it means at all that she isn't cut out for competitive gymnastics. I definitely agree with Dunno that there are a LOT of different personalities and levels of maturity on any gymnastics team. So be very wary of a coach / gym situation that doesn't accept these differences. I'm not a fan of the gymnast as a machine mode of thinking.
Best Wishes,
ZZ Mom