I am going to be VERY honest here, but here are my thoughts, from someone who is only going by what is in this thread:
-Five privates on top of 20 hours is absolutely insane. If five privates are needed after 20 hours of training, the training is either lacking or you are pushing her WAY to hard. Even if your daughter is begging to go in the gym, your job as a parent is to ensure she doesn't burn her self out, open herself to repetetive motion injuries, etc.
-If everything you say the coaches is accurate, that gym is absolutely toxic and you should not allow your daughter to be in that situation 25 hrs a week!
-And finally....I am getting vibes (from your posts and responses) that some of this toxicity might be coming from your side. The tone of your posts and your descriptions of the other girls and parents make me think that we would hear a very different story from the other parents. Might be time to take a step back and see how you are acting with, and seen by, other parents in the gym. If you are getting any favortism from the head coach, it is most likely because of $$$ from privates ....any coach that was truly focused on the long term success of their athlete would not put them through 5 privates a week.