Privates should not even be on the table for ‘uptraining’ at this age. Has the coach expressed that she seems disinterested? If so, maybe she needs to practice up a level once meet season is over to keep her more engaged. This is what off-season training is for at our gym. Our off season is during summer break and usually involves training up a level practice wise with a full week of team camp at some point. Teams are broken up and each have a week where they spend 8-12 hours a day with their cohort doing some intense uptraining with guest coaches that specialize in dance, yoga, etc. Ultimately the owner/ coaches make a decision by the start of the next school year as to what level your kid will compete and practice at for next season based on the progression they make during summer session. IMO, kids should only be getting privates at her age/level if they are needed for learning new routines/ choreography for optional levels where each person’s routines are individualized. They can also be used as a tool for helping your kid get a specific skill who are so close to having it and need it for the level that they want to compete on (for example a BHS for L3), or maybe for an extra pick me up after a not so great meet with 1:1 attention from a coach. There are gyms that will really push privates on you, and IMO, they should be able to do everything your kid needs during the hours they spend at practice, including uptraining. If they are using privates as the only means for uptraining, I would honesty be looking into other gyms and their program structure. There are some gyms that have very large teams and really can’t focus on uptraining because of the size of their practice groups. They focus on the task at hand, which in lower level compulsories is unfortunately repetition and perfection of pre-set routines with pre-set and built in deductions. What is the coach to gymnast ratio? If you have a larger ratio (personally, I would consider a 1:7 or less as optimal)- maybe your DD needs a gym with smaller groups and opportunity to have built in uptraining.
For reference, I have an 11yo L7 who has only ever had 2 privates. One was to learn a new choreographed routine and the private lesson was included in the choreographer’s overall cost. The second one was after a really bad (for her) meet on bars and needed a pick me up with one of her favorite coaches. I also have a 9yo XG who has never had or needed a private. It is not common for them at our gym because everything can be done during practices and as explained above.