I really dislike this, especially because it ultimately results in worse care for the kids.
I think the place I take my kid to for therapy is likely private equity owned, but it's also about the only place available in my area.
I know they are charging around $80 per season, and I also know that the salaries for their therapists are around $25 to $30 per hour.
That leaves a very healthy margin for everything from employee benefits to building rental to admin (which I think is probably where the majority of margin goes).
It's worth asking if you're curious or have a good relationship with the provider. Some may be willing to tell all about their ownership or management situation if prompted politely.
I know of a childcare center that was acquired recently. Not sure if by PE or just another business, but the employees are less than thrilled with the changes.
I think you're over-estimating how much slack there is in the organization.
A rule of thumb is that benefits cost an employer 25-30% of salary. So you're already pushing to 50% of revenue going to direct salary costs. Then there are employees in non-revenue roles (HR, legal, accounting, IT, etc...) and employees doing non-revenue work.
Finally, you have rent, licensing, insurance, and all the other fixed costs.