In the US, "health insurance" mostly isn't about actual insurance, it's just part of the evolved process of obtaining care with a degree of oversight and (semi-)centralized records. You can pay out-of-pocket, in much the same way that you can represent yourself in court: nobody can deny you the option, but the system isn't really set up to operate that way and something will probably go wrong when you try it.

That's not entirely true. When I was uninsured, I was able to negotiate significant cash discounts with many out-patient providers. They had a very good idea what the overhead of billing an insurance provider was and were willing to lop that off their charge.

Where it gets tricky is with in-patient care. There I would agree that it's more like representing yourself in court.