I'm mostly familiar with the UK system, but medical professionals make pretty much all the decisions here, with a large degree of discretion according to their professional judgement (and they never have to adjust or delay their care based on whether you can pay). Except for some particularly expensive treatments (think CAR-T for cancer) which are not available at all in the state funded system. But you can still pay for those privately if you want to.
We could just not do that. If you change the flow of control certain problems solve themselves. Think about a landscape where government funding multiplies the patient dollar, for example.
Both have similar health care outcomes - they have ready access to quality care, specialists, etc. ER/A&E is available. The biggest difference is the perceived cost and stress incurred by that cost. My uncle doesn't give much thought to health care - he can work, retire, whatever and be assured a reasonable level of care. My BIL will work to 65 or beyond, fighting red-tape the entire time, then retire and still have to deal with supplemental programs.
Looking at another uncle, who was a small business owner in Scotland vs my father (also small business owner), it's similar to above, just with more money at stake. Uncle also purchased additional insurance on top of NHS for faster access to selective care, still cost less than insurance in the US, even after accounting for tax differences.
American's kid themselves when they say the Western Europe has higher taxes. Once you account for medical care, college funding, and other similar things, it's pretty close.
Nothing's perfect, but the plan differences seem stark. For example, my wife had a crappy marketplace plan and I had a plan through my employer. For her, an MRI was denied, denied, then finally approved with many calls. For me, it was approved immediately. For her, pre-auth to a specialist was denied until her doctor went and tried a different referral strategy. For me...well, I haven't been denied yet. It goes on - same city, same hospital, some of the same referrals, etc.
I've come to think the price discrimination really does mean we have class-based care which seems to allow for the sensationalism. Combine a dire scenario with a working or indigent class American, and they don't have to exaggerate much at all.
It does make a big difference exactly where you are in the US, however. Some places have a glut of healthcare providers and other places don't.
Where in the US did you have to wait months? There seems to be an MRI/imaging location in every other shopping center in the US right now. I've never had a problem getting a same day MRI when needed. Perhaps you were waiting for the 'free' one your insurance would accept?
Now try to schedule a colonoscopy. It'll probably take two or three months.
This happened to us with private healthcare. There is basically one specialty group for the procedure my family member needed so any 2nd opinion request just got routed back to the same doctor, "Oh, your Dr X's patient". Also, we could barely afford the procedure so we missed out on some follow up testing that would have verified things worked properly and basically got blacklisted from that practice so hopefully it's resolved...
I'm not sure how the other Nordic countries do it but I think it's probably similar.