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.