I wouldn't go that far. But it is true that there are far many more possible outcomes because of how diverse the private insurance system is.
At the very least one guaranteed difference is that you would NOT not get a bill.
Then, a common scenario is a health insurance setup that covers 100% of the cost beyond a threshold, while the individual is responsible for the cost below that threshold.
My threshold is $6k for example. In the above scenario with the elbow, I estimate that the total cost would be around $100-200k. So I would end up paying the full $6k, and the insurance would take care of the rest.
The other common scenario is one where the threshold is much lower (e.g. $200) but the insurance covers e.g. 90% of the total cost. I find that option more risky, because I know the total costs can be staggeringly high, and 10% of a staggering number is usually also a staggering number :-)