I feel like the >(cost!!) factor can't really be overstated there. IF you have money then you can get healthcare, if you don't, then you're out of luck. It's a massive gate which needs to be passed, and if you can't pass, then you're locked out with no alternative.
That doesn't sound great to me (read: broken), because it means its not evenly accessible across your society. I wouldn't count that as a good system to be honest.
I have limited experience with the New Zealand and Japanese medical systems, and from a similarly arms-length engagement with them, I think I would prefer them over everything I've read about the US system. You still have the same issues with having to advocate for yourself, but you can at least get access to good healthcare across the board - and if you go private (which as a late 30s / early 40s guy, cost around NZ$60 / month), you can get access to care at private facilities much faster than via public.
It is most definitely imperfect, but it _is_ designed to be accessible to everyone.
In theory, Serbian public healthcare system is available to everyone, but in practice, wait times are extremely long (you might wait 6 months for an MRI for a suspected cancer by a specialist; specialists are booked a year in advance sometimes, though GPs are pretty available).
The way system (and patients) survive is by relying on "connections": someone you know gets you to "jump the line" (it could be a nurse or a doctor or even a janitor at the same hospital you want to get a specialist to see you).
It ends up being similarly "fair" (iow, not), but at the very least, everybody will prioritize kids and young adults (though in children's hospitals, same pattern is present).
Private healthcare is reasonably accessible, but best experts are with public healthecare institutions (uni clinics) or when working externally, booked 3-6 months in advance.