I'm sorry, but to me these still all sound like reasons one would deep down want to leave a current gym and go to another. It's one thing to desire this extra stuff from say a nationally renowned gym like a WOGA or Chows (which are what camps are for), but when it's a private lesson w/ a rival gym w/in driving distance, you're clearly unsatisfied even if you don't realize it yet IMO.
Perhaps if all/multiple of the reasons cited for desiring privates were true at the same time (better equipment, expertise, timeslot..), that would tend to indicate a deeper dissatisfaction, yes.
The more "decorated" gymnasts (e.g., TOPS, Future Stars, other elite camps) who earn special clinics and camps take advantage of a range of different coaches and their unique teaching styles pretty frequently. Coaches each contribute their experience and pointers in the best interest of the athletes (or so I assume - I have only been to one such clinic with my child, but coaches did seem to all genuinely be trying to help all kids, including 'rival' gym kids).
Is it fundamentally "wrong" to think an occasional dose of an an additional expert helping your child might be beneficial? Does it really have to be a person from Chows or WOGA (or similar big name) to be potentially helpful? Does it really mean that a person is "clearly unsatisfied" and disloyal to even explore the thought of whether some targeted work with an additional expert might be beneficial? Not everyone's child has access to these experiences (open camps perhaps, but that may or may not be a fit), but could reasonably wonder if exposure to more/different coaches could in some way help her child, too, since it seems to be such a common experience to "cross-coach" in the clinic atmosphere.
I realize privates with another coach could be counter-productive for other reasons stated (different training plan, short-term vs. long-term progress), and so I'm certainly not advocating this should be the "norm" at all(!!), but I would not assume a parent who is wondering out loud if "another set of eyes" might help her child is automatically dissatisfied and disloyal.
Our coach is, imo, one of the most technically proficient, hard-working, kind, and trustworthy coaches in the whole region. I 100% don't want to go elsewhere. That doesn't mean that I would not welcome another coach's time and input to work with my son if it were offered, available, and welcomed by our coach. "Rival" is a construct separate from the best interest of an individual child's gymnastics.