Let me wade in ... I don't know the coach, and I have no way of commenting on her professionalism, ethics, etc. However, there is no coach in the world who can identify that a very young, beginning skater has an above-average natural talent in skating, especially not at age 5. You can see look-fors, which are a range of things which include:
Balance
Coordination
Interest
Presentation ability
Flexibilty
Body type
A 5 year old is still a very small child, and you cannot tell at that stage what body type s/he will end up with, nor can you really measure flexibility, etc. Again, I cannot comment on your coach, but, I would be wary of any coach who is telling parents that their child is a "natural" at that age - maybe it's just a great deal of enthusiasm, but, realistically - they can't know. Nobody can.
Your child is a child. Skating should be just - fun - at this stage. If s/he wants private lessons - hmmm - my DD was in privates at age 5, and for a similar reason, that group lessons drove she and I both nuts for the reasons you cite, and, as the parent of two older skaters, I am picky as all get out in terms of quality of edges, turns, etc. However, we also picked a coach who was focussed on keeping it fun, unpressured, and who had a similar philosophy as we had - that kids should be allowed to progress at their own rates.
I would ask your DD if she wants the privates because she really loves SKATING or if she just likes doing one-on-one with an adult (some kids enjoy that very much). That will help you in understanding what is going on. Also, 5 year olds are very easily influenced; if they get the idea that they will be "amazing" if they do lessons with a particular coach (again, your coach may not be doing this, but, I have seen coaches spin little minds this way in their eagerness to get new students) - so, an open and unstructured conversation with your DD about why she wants this, and where she got the idea would be good.
I would also say that your DD is skating too much for a 5 year old. That is a lot of ice time, and a lot of lessons.
Sigh. Can you tell I'm cynical? Sorry ... I'm just very protective of little skaters and their parents - everyone should have a good experience in this sport from day one, if possible.