However Healthcare in the US isn't a regulated monopoly like that.
I think it was this one
I'll argue whatever PG&E problems are they don't rise at all to the level of healthcare. And their problems are due to poor regulation and the US's current terrible management culture.
This works in Denmark because of Denmark, but it's still a system that has obvious weakness.
[1]: https://en.wikipedia.org/wiki/Corruption_Perceptions_Index
Essentially, the state (or region) contracts a company to deliver ambulance services. The contracts usually last a few years and are limited to a region.
Furthermore, the company that wins the contract (usually lowest bidder) is held accountable is too many responses are too slow, etc.. usually with hefty fines.
I guess you can call it privatization, but it's more of a regulated monopoly. Ultimately, these things outsources uncertainty from the state to the private sector. I'm not convinced it's always a good idea, since companies can just factor the cost of fines into the contract... Granted if there is competition, it can drive the prices down a LOT.