"Someone with no money or bank account should be able to receive non-emergency care without needing to pay."

This muddies the water on this market quite a bit. Many countries (even America!) have free healthcare for the truly destitute.

Absent that I would be putting manna on NO here because I think it is very likely that additional fees are introduced (eg a charge for missed appointments, the option to co-pay for weight loss injections for those not otherwise eligible, etc) but I think that people with "no money in their pocket" will be exempted from these charges. It will be those with "some money in their pocket" who are hit with these.

Are there any developed countries that would resolve no for this ?

I am assuming you don't count National Insurance tax contributions?

@CromlynGames You assume right - I would not count this.

what does "at the point of use" mean ?

@Odoacre I will use the definition currently on wikipedia, which was what I already had on mind:

"The vast majority of NHS services are free at the point of use.

This means that people generally do not pay anything for their doctor visits, nursing services, surgical procedures or appliances, consumables such as medications and bandages, plasters, medical tests, and investigations, x-rays, CT or MRI scans or other diagnostic services. Hospital inpatient and outpatient services are free, both medical and mental health services. Funding for these services is provided through general taxation and not a specific tax."

I have made an equivalent market for 2035: