Because it's not everyone paying for everyone's health care, it's one entity paying for everyone's health care (using funds they get from everyone).

That one entity can unilaterally refuse to do business with abusive vendors. They can negotiate for lower prices. They can notice and investigate discrepancies in pricing between similar cases. They are in a position to judge which types of care are and are not cost effective and can refuse to cover the latter.

So that explains why having an insurance company helps, but it doesn't explain why a private insurance company can't perform the same role. For that, we have to introspect a little into how insurance firms work.

An insurance company is, in essence, a mechanism for transferring funds from lucky people to unlucky ones. If your car gets totaled then, in essence, a bunch of other people with working vehicles have all volunteered to chip in a couple of bucks toward buying you a new one. The insurer is mostly facilitating the transfer and skimming just enough off in the process to cover their administrative costs.

The private insurance market works well enough for most types of property because the volunteers can't predict in advance whether they're going to be lucky or not and because there's a realistic upper bound on how much money the unlucky ones will need to be made whole. Neither of these assumptions are true for healthcare. Many people were born into pre-crashed cars, to stretch the metaphor, and at least for the foreseeable future it's not possible to decide to replace a body instead of attempting to repair it.

So we need* some non-market mechanism to incentivize or compel insurance companies to cover the already unlucky: people with pre-existing conditions. But that introduces a new problem. The lucky people are being asked to volunteer to chip in more, and at least some of them will stop volunteering. That causes the prices to go up even more, which causes a few more volunteers to leave, which causes the prices to go up again, and so on. Pretty soon there's no volunteers left to transfer funds from.

So we also need* some non-market mechanism to incentivize or compel healthy people to contribute to our insurance system. And with those two requirements (needs to cover everyone and everyone needs to contribute) we are left with a problem space that the State is really the only entity with the ability to implement a solution. It doesn't have to be socialized medicine, per se, but it does basically have to be big government of one stripe or another.

\* Yes, we do technically have the option of compelling innocent people to suffer and die from otherwise treatable diseases that they're not wealthy enough to afford to cover out of their own pocket. If you're aware of that fact and comfortable with it then okay, but please do understand that it is the societal tradeoff that we would have to make to avoid some sort of serious state involvement in provisioning medical care.