By definition a free market in healthcare requires forcing some people to die from a treatable injury or condition to discover the optimal price. That's part of the definition.

Because the supply is constrained I feel like that's the case regardless of how the system is financed. (There are only so many providers, facilities, etc.)

It seems like the core problems with US "system" come from having a "shadow" socialized system (by way of EMTALA[0]) that offloads costs onto the "free market" (read: patients or insurers who can pay) and a shocking inability to have a mature public conversation about how care will be rationed. (That conversation still happens but it's between providers and insurers and not subject to public scrutiny.)

[0] https://en.wikipedia.org/wiki/Emergency_Medical_Treatment_an...

Only if free means pure or academic here.

Free market usefully means both sides can choose to participate in transactions which means price is how they do that. It is a useful concept but it doesn't cover everything, like what you mentioned and things like if emergency patients are shopping around hospitals or if the average consumer is capable of being well informed in such a market.