My practice goes the other way. We are private but work most of our days at a hospital. We bill our own professional fee, but the hospital provides all staff, referrals, supplies, support, etc. Hospital doesn't control how much or what days we work. We limit ourselves to 4/hour, 24 max per day, with a dedicated lunch from 12-1 (often 11:30 to 1). Partners work 3-4 days per week. If our new patient waiting list gets more than a week, we pick up days to see them. If our procedure appointments get too far out, we add a day.
It's a good work/life balance and while we may not make the huge bills others are, we make plenty of money.