Our preference is Federal (COE, NFS, BLM, NPS), State, County/City. We have only stayed at private campgrounds twice and only because of location. We don't use what most people call "the amenities" and for us, amenities are large spaces and privacy.
For us, the priority is usually the view through the back window. If all we can see is inside our neighbor's window, we might as well be back in the city.
I think it all really depends on why one is 'camping' in the first place. If, at home, you don't have access to a pool, gym, or hot tub, those will be attractive draws. If you don't have access to space and privacy, those will be attractive draws. 'Wanting to get away' means different things for different people.