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: