Figures on mortality in hospitals have been looked at, at least at a state level, for years. Hospitals in most states (whether private or public) have to report fairly comprehensive data to their state government every month. When I worked for a provider of patient management software, Victoria collected more data than other states, so nationally comparable figures might have been lacking.

They've looked at the data, seen that public hospitals treat sicker patients, and ignored this metric as useless, long ago. (10-15 years ago!) Small public hospitals tend to be rural and be a dumping ground for patients who have nowhere else to go, due to lack of alternative services. (But the "no private hospitals in rural areas" claim is exaggerated too.)