Although I had private insurance with my first baby, I decided not to use it. Obstetricians in private system would be paid out of my pocket anyway (well, mostly) and it turned out it would cost me really big. Not to mention that you need to book one early. Same goes for scan, if you aim for public, secure appointment now or as early as possible. I had to go private cause waiting time would see me pass my 12-14 weeks term anyway.
So, I went public (health) - except scans, as I stuck to the same provider with all my 3 boys (I decided that if I trust them and they are good and thorough, it's worth it). And health care was good, birth and stay in the hospital also good/very good, midwifes were really great and supportive, and I felt really well taken care of.
Having say that, I had low risk pregnancies, even considering my age, only that second and third was 10 days overdue.