Private insurance is very good at effectively pricing risk. Bad drivers should pay more for car insurance, and homes should be priced out of being built in areas with pre-existing risk factors like being below a mountain prone to mudslides
http://www.pendinghorizon.com/2017/09/private-insurance-is-about-making.html

But when it comes to healthcare, we decided that the pre-ACA status quo of insurers pricing health risk by charging more (or denying coverage) for people with pre-existing conditions was immoral. So we effectively regulated away private insurers main task

That left private health insurers with three remaining purposes:

1. Spreading/pooling risk
2. Negotiating prices
3. Customer service

They are very bad at all three

When it comes to pooling risk, they are actively lobbying the government to help do their job for them, with a re-insurance program, because they can't effectively pool risk themselves. A single teen nearly took down the entire Iowa individual market https://www.desmoinesregister.com/story/news/health/2017/05/31/hemophilia-patient-costing-iowa-insurer-1-million-per-month/356179001/

When it comes to negotiating prices, insurers have failed spectacularly to keep prices under control, with little bargaining power against quickly consolidating hospital monopolies. Many have proposed doing all-payer rate setting, which would take this job from them as well

And when it comes to customer service, their customer satisfaction ratings are consistently below public insurance, and that may be in their interest because they make more money the harder it is for people to utilize care

So unless we want to use private insurers to restrict access, or to provide rich people better care than poor people (neither of which seem desirable to me), I just don't see what value private insurers are bringing to the table