You’re first sentence is the correct one. We do have an NHS equivalent (called Medicare), but this was likely done by a private surgeon, if indeed it was done in Australia at all, given how many of these athletes live overseas anyway…

I can think of maybe six or seven large private health insurers off the top of my head, but I imagine there’s more than that… It tends to be a state-by-state thing, though, so I know that two of the insurers where I am only operate here…