Now that I have that out of my system.
I really think that privates should be used to complement regular training, they should not be part of a regular training schedule. I think a child is better off in a group of their peers, where they will learn by watching and doing.
I think three level 4 classes a week is plenty for a six year old, she is very young and should be doing other things besides gym. I would really wonder about any coach that wants a 6 year old in the gym five days a week.
Gym at any age, and especially as this young age, should be fun and should be with other gymnasts.
I have two DD's in gym, my nine year old, level4/5 (different system here) has never had a private and probably never will. Older DD, 11 years about level6/7, has had about 7 privates in the last two years. She had them as she had to miss so much gym due to injury, she did them right before meets to get her routines down.
I would keep your money in your wallet, you are going to need it later on if she sticks with gym, it gets very expensive as they move onto team. Check out this thread in the parents forum.
Is it worth it?....opinions needed
In the end you should do what you think is right for your child. Remember that out of every 100 gymnasts who begin at level 4 only 50 of them are left by levels 6 & 7.