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.
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.
That's just corruption. That's not equitable. In the UK that would be criminal and there's pretty much no way to game the system. That doesn't mean there aren't queues, but nobody gets a preference.
But I did hear good things about UK public health system as being one of the best working public healthcare systems, even if no system is perfect.
In the rest of the world, it's controlled by extended wait times. In the US, it's cost. You can have moral arguments about what's fair, but that's the economic reality.
At the time I as studying this (early 2010's), the average wait time of diagnosis to first treatment for breast cancer in the US was 12 days. In NIH, it was 5 months.
You read that where? What evidence do you have? In my experience, Americans are often blissfully unaware about how poor their health services are.
As someone who has seen many health services up close (from founding a health tech company and literally visiting myriad health establishments around the world, over the past few decades), the US hospitals and clinics always struck me as the most run down. But as soon as I walked in the boardroom at UHG or HCA, it suddenly all became a lot more glamorous.
And whilst the US does ok with cancer outcomes, so do other nations like Germany, Switzerland, Japan, South Korea, Singapore, Canada, and the United Kingdom.
Most advanced nations have universal healthcare with similar outcomes to the US, but for significantly less money. They also don’t leave citizens destitute for having the unmitigated gall to impose an unexpected health emergency on themselves.
The US system funnels profit, it doesn’t provide better healthcare.