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...
It is disheartening to see anyone in any system suffer. I am quite sure if I was someone suffering in your example's case it would be hard not to be angry at the system. When it's you hurting you don't care so much if a system if overall better, better for others, or was even better for you 99/100 times thus far, prevents a bigger problem later etc...you just want to stop hurting.
But an acquaintance with Canadian relatives told me he knows a middle aged single man that was denied a knee replacement he needed because of these trade-offs. Left with decades to live, partially disabled, because: "Old? No family? Well fuck that guy then."