Private health insurance companies don't do any of the things you just said. They don't deliver efficiently, or sometimes at all. They are not affordable. They don't allocate medical resources effectively. In fact, historically they have limited care (pre-existing conditions, lifetime maximums, inconsistent reimbursements for procedures), denied claims, and left millions uncovered. All of this while raking in enormous profits for themselves for doing paperwork.

If someone were designing an insurance system from scratch they would never in a million years pick ours. You know what is popular though? Medicare. Canadian Health Insurance. French and German single-payer. Hell, even the damned British NHS, which I would never advocate for the US, is more effective!