Public healthcare incentivizes the government to protect people's health; the American system incentivizes politicians to protect the profit margins of the insurance industry.
I'm not sure if you have the law in Sweden, but here in Norway we got a "Smoking law" that prohibits smoking in public places to promote people's health and reduce the cost of running the government hospitals.
To be fair, the smoking laws here in Norway are a little more relaxed than they were when i was living in the US. Want to smoke in the hospital parking lot? You'll get escorted off. Same for areas outside some airports, but I can smoke outside the airport here in Trondheim before I get on the bus. No real problem smoking on the sidewalks downtown. Some ask you to go away from the doors, but that isn't a real issue and I find it reasonable. Companies aren't firing people for smoking, and I don't find they tell folks they simply cannot smoke during the work day (including lunch).
You can smoke in dedicated smoking areas outside, but it's actually not allowed to smoke on bus-/train stops/sidewalks etc. There are just no one to enforce it.
I guess the places where you get escorted from in the US are privately owned and not due to a law (but I could be wrong).
The companies firing you for smoking might also have to do with a bit stronger labor laws in Norway :)
This is one case where the libertarian argument that governments are fundamentally coercive works against you. Insurance companies can't take such broad actions to protect health as government can -- things like taxes on products with negative health externalities, regulations ensuring workplace safety, etc.
The argument here was about incentives -- i.e. the practical component of how our laws get updated, not about the theoretical component. Your original argument was that private companies are incentivized to pursue safety, and my counterargument is that incentivized private companies do not nearly as much power to effectively pursue safety as incentivized governments do.
There's a crop of startups now that are incentivizing healthier lifestyles by gamifying healthy activities, routine doctor visits, etc. Insurance companies pay them per head to make their software available to customers, and reward customers with the most health points with lower premiums and such.
Insurance companies, by and large, make their money on the delta between the prices they charge and the costs to deliver service. If they can delivery service to a predominately healthier population, then they can deliver service for lower costs, meaning they can lower costs to be more competitive, while being able to make more profits and deliver better outcomes.
The notion that they want to drive up healthcare costs is kind of ridiculous. If that were true, they'd be rushing to sign up patients with terminal illnesses and costly pre-existing conditions, but the fact of the matter is that the government had to literally force them to for that to happen.
Taking care people and difficult medical conditions will likely never be dirt cheap, right? Unless you're envisioning Dark Mirror style robot care and cure everything pills.
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:
