One thing that's often overlooked is the cost of administering a billing system. If you're going to distinguish between the insured and uninsured you need a system that keeps track of who has insurance. You need paperwork (or digital equivalent). You need accounts payable staff to send invoices and chase up unpaid bills. It can, counter-intuitively, work out cheaper just not to charge anyone.
That is not required complexity. You can just send the bill to the customer.
Who is the “you” in this scenario?
The company offering the ambulance surface. Similar to how a pool cleaning company will send you a bill if you call them and have them clean your pool.