I recall a discussion with a plastic surgeon once. He was an example of someone inside and outside the medical system. He could charge a fee (no government reimbursement) for vanity medical procedures. For example a tummy tuck (remove excess skin so it does flop around like Richard on that early episode of survivor ) the government covers the procedure in hospital if the patient lost 100lb or more (or whatever th metric equivalent would be). For less, it was an outpatient procedure in a day clinic, no government-funded hospital involved. There’s a particular condition where the ears stick out significantly, that is covered (apparently kids with that problem can experience severe emotional trauma from peers). Obviously, plastic surgery from incidents like burn trauma. But nose job? Face lift? you pay.
Also, procedures have to be approved and accepted medical practice. For example, there was that treatment for MS that involved widening the neck artery (angioplasty or stent). Assorted provinces funded tests where some patients got treatment, but a doctor could not simply do the unproven treatment and bill the province. (I knew at least one person who travelled to Poland to have it done. It helped for a while, he said.) When tests seemed to show it was not effective, it was not added to the approved treatments.
An interesting issue occurs when Big Pharma produces a drug that costs a small fortune - especially if, like the recent Alzheimer’s drug in the news, it fails to show significant (or any) value. Provinces will often decline to add that treatment to the covered procedures list.
I’d argue that St. Jude’s isn’t necessary from a FUNDRAISING standpoint - but even in places with universal healthcare. there are going to be places that specialize in certain things - as St. Jude’s does with children’s care.
Which is, of course, a slightly different question than was raised by the OP.
Ronald McDonald Houses, and the services provided the Shriners (physio, wheelchairs etc.) would seem to be necessary almost anywhere, to help fill in the gaps for things that a health system cannot really provide. I know little about the Shriners; rather more about RMcD Houses - having done volunteer work with young kids whose families were in the exact situation back when the local hospital (NC Memorial) did not have one. The hospital staff supported families as much as they could - but simply did not have the resources. There are locations in Canada and the UK. and I’d bet in other countries with universal healcare.