This highlights the problem with privatizing things like healthcare and education, something libertarians don't understand. It works for the Koch's because they can pay for anything. It doesn't work if you're not rich.
I wouldn't say privatizing is the problem. It isn't. Private is often, or generally, good, as it gives you the freedom to pursue good ends without unnecessary involvement of state bureaucracy. It's bad and weird to have the state involved in everything. It's for-profit that is problematic in the mentioned cases.
Healthcare, insurance, banking, education, and so on should be not-for-profits or nonprofits (depending on the case).
All nonprofit means is that you are not organized with profit as a primary goal. It doesn't mean you don't make money, and it doesn't mean that executives don't have outlandish compensation.
Main problem with non profits in my understanding is, that they are often created for tax evasion purposes, but the legit non profits still get the regulatory heat.
What is the incentive for a private entity to engage in non-profit business.. charity?
Nobody want's the state involved because they think they'll do a better job, they want the state involved because it's the last option available with incentives remotely aligned with the benefit of the polity.
Healthcare, insurance, banking, education, and so on should be not-for-profits or nonprofits (depending on the case).