We have the same in my country.
Private hospitals can be good but medical aid is expensive and only a few can afford it.
Public hospitals have over worked and disgruntled staff. And doctors are leaving because their salary wont change.
Not the case at all over here. "public" hospitals are usually the hospitals that are also part of university complexes. They'll generally have the best doctors, many times even world-renown in their area of expertise.
Next to the hospital work, they'll oftenly also have their own private practice where they do consultations and stuff.
So the general rule over here kind of is:
If you want a hospital with okay / mediocre care where you feel like you are sleeping in a 5-star hotel, then go to a private hospital.
If you want a hospital with the best care and the best doctors, you go to a public hospital, swallow your snobbish pride and accept that you might share a room with a random citizen of any of the many walks of life.
I'll choose the best care any time. Regardless of being able to pay the 5-star hotel or not.