South Africa is the only place in the world where private sector healthcare is better than government  - so you can't from this extrapolate anything, even about the rest of South Africa. There is an obvious cause for South Africa's public healthcare being worse-  that cause is corruption, but since corruption happens in both sectors - it could just as easily happen in the private sector.
Good luck withholding payment by the way. Your medical aid generally doesn't let you choose which bills it will pay. Meantime the largest medical aid in the country is also teh worst one in the country  -so how are they so big ? Well they are very, very good for employers. You're not even the customer - they feel bugger-all for you. If you need something big, and they can find any way whatsoever to avoid paying, they will find it. Tying medical access to employment is, by itself, an evil outcome that harms the entire economy. It stops people opening new businesses for one thing.