It depends.
A university hospital would in my opinion be the best option. All of them are public.
There are, however, only a few of these.
Having said that, I have health insurance to cover some private hospitals.
I used to have cover for all hospitals but the real economics snuck in and and I had to cut my cloth etc., I suppose, like most other people.
I do recall that I bumped a male patient out of a male surgical private room a number of years ago in a public hospital because my insurance plan was higher than his. He was not not a happy camper but I was too sick at the time to care.
Marion