Kinda works that way under the NHS in the UK too. Eye and dental care are handled by private opticians and dentists.
You can get financial assistance but most people don’t qualify. Working people generally have to pay for both themselves.
Also you only get medication paid for if it’s a long term illness or you’re on financial assistance.
Some hospitals used to have emergency dentists. Not sure if some still do but I know my local one doesn’t.
Still heavily subsidised, it’s not like we are paying “full price”.
NHS dental charges
There are 3 NHS charge bands: Band 1: £25.80
Covers an examination, diagnosis and advice. If necessary, it also includes X-rays, a scale and polish (if clinically needed), and planning for further treatment.
Band 2: £70.70
Covers all treatment included in Band 1, plus additional treatment, such as fillings, root canal treatment and removing teeth (extractions).
Band 3: £306.80
Covers all treatment included in Bands 1 and 2, plus more complex procedures, such as crowns, dentures and bridges.
https://www.nhs.uk/nhs-services/dentists/dental-costs/how-much-will-i-pay-for-nhs-dental-treatment/
And it has greatly improved British teeth. The old stereotype of bad British teeth was because of how things were before the NHS. My English father’s teeth were so bad that he ended up having to go to Costa Rica to get the surgery done because it would have cost $10,000 in the U.S. out of pocket. He was born in 1931, so the NHS didn’t even exist until he was 17, but he always considered it the greatest accomplishment that Britain had achieved (my father was also a die-hard socialist who bragged about how he booed Churchill when Churchill visited his high school). He’d be pretty appalled at the state of it now.