I'm not sure anyone has solved the problem, though.
"For profit" isn't really the problem, fee for service is.
Value based arrangements and capitated payment structures are real improvements.
When you don't have any market forces constraining utilization, you get massive access problems.
The United States has both market forces and massive access problems. I'm not sure you can establish that causal relationship.
I don’t think it’s useful to categorise these things as “solved” or “not solved”. They always exist on a scale.
I can tell you I am infinitely more happy living with the Australian medical system than what the US has even if it isn’t flawless with zero issues left to deal with.