New Zealander here, and I will speak about cases I have direct experience of (and project a little based on what I've read in media).
Firstly, our social healthcare is actually pretty good for emergency procedures.
But there is only so much money budgeted per year for healthcare, and everyone in need competes for that limited pool, so there is a variety of ways care can be delayed or denied.
"Elective" procedures like hip and knee replacements can have long delays (queues) and seemingly odd requirements. One friend was "too" young: they had to wait for years in severe pain because they "shouldn't" need their hip replaced yet. Another acquantance had a record of hurting their knee as a young child: so they were not covered by ACC for knee replacement even though it was a ridiculous excuse in circumstances. Once a procedure is approved, you get reasonable treatment, and it is free.
There is a certain amount of gaming of the system - if you can find a sympathetic specialist you can jump the queue a bit. However that is a zero sum game, and it benefits those with time, intelligence and contacts (and screws some minorities).
Pharmac has a yearly government budget, and they choose which treatments can be afforded. We do the same with roading - cost benefit analysis is done for all roadworks and the best get funded until the fixed budget is met (small country, so one main roading department).
It is interesting that our two biggest healthcare insurers (maybe covering 80% of private insurance) are non-profit co-operatives: all premiums are either paid out in care or pay administration overhead. I think overhead works out to be about 10%. Fantastic system: pay premiums and don't get sick is winning, getting payouts is often actually losing. It's like a socialist system for everyone that has some disposable income. Private healthcare premiums are extremely cheap compared to the US e.g. "the premium for a member aged 0-21 years on our most popular plan, Wellbeing Two, is just $9 a week", and "Pay for 2 kids, the rest are free*"; you can see what is covered by that plan here: https://www.southerncross.co.nz/Portals/0/Society/EFulfillme...