I work in a practice of all CRNP's and it's great. We have a dr. who is our "medical director" but she never sets foot in our office and doesn't have anything to do with our practice (other than the check she collects each month!). She is not required to co-sign anything or oversee our patients. And she's not needed to do that. The NP's are highly trained and they don't just do the "simple" cases....how insulting to say that. They do everything the doctor does, but the main advantage (our patients say anyway) is that they take more time and listen better. If our NP feels they would benefit from a specialist, she'll send them out, just like the doctor would. In fact, sometimes she is quicker to do that than some doctors who feel they can handle everything. Sometimes we have patients who question the difference between a doctor and an NP and I have a difficult time answering that because for all practical purposes there isn't much difference. I'm sure that's not the case everywhere, it will depend on the various skill sets each possess.
Oh, and our NP's are all women and we've had maybe a handful of adolescent boys and men who aren't comfortable with that. I respect that but most are able to be mature enough to get past it. Personally I feel like it's payback for all the years I had to see a man gyn!
As far as the original question, if you are scheduling your appointments far enough out I don't understand why you shouldn't be able to ask for the practitioner you prefer.

If it's a sick visit, you take whoever's available, but you should be able to book with your preferred provider for an appt that is weeks later. If they don't allow that, I'd be moving on if seeing the ped is important to you.