The state governments don't actively chase private insurance...

The state governments don't actively chase private insurance holders to attend public hospitals but the hospitals, with government encouragement, do pressure you to be treated as a private patient once you are there. You usually get nothing extra for it but they get most of their costs picked up by the private insurers. Over the past four years, a family member of mine has twice been admitted to public hospitals in an emergency and her private insurer ended up paying, with the public hospital out of pocket for just her excess. This doesn't affect the private hospitals directly as the emergency patient would not have gone to them anyway, but they do end up having the squeeze put on them by the insurers as they endeavour to maintain profitability.