In Australia, we have a public-private model for hospitals and health services. However, I think there's starting to be a greater awareness that private health insurance for many holds increasingly poor value, compared to what is available publicly under government funding as well as simply saving up what would be paid to private health insurance companies and instead directly pay health providers from one's own bank account. To date, everything I've learned about private health is that it's to maximise profit while being able to provide a semblence of good value healthcare.
Here are a some links I've collected on the topic - obviously, I'm a little biased but I have yet to see any compelling case for privatised healthcare on essential services: