The existence of private insurance strongly correlates with higher health care costs

[Recursion]

11. I'm not talking about Medicare advantage

I'm talking about Medicare. It uses private insurers to provision actual care. Medicaid does even more.

but there is no accounting for the massive administrative costs of private vs public insurance.

Right, but Medicare's low overhead rate only counts getting the money to the provisioning insurer. They take premiums and subsidies and give money to Blue Cross and a few other big provisioners. This takes very little overhead. The provisioners then do the actual billing and payment work, which brings me to:

just what do these companies bring to the table?

Provisioning. Something Medicare doesn't even try to do on its own.