Preventative care. If you have insurance that covers checkups, screenings, etc. then you get that benefit. If you don't have the insurance and can't afford the out of pocket expense, you skip. The issue is that then people wait until they're in really bad shape before seeking treatment meaning that outcomes are worse and treatment is much more expensive than if the illness had been caught earlier. Who pays for that? We all do through increased premiums.
This doesn't happen in a well-run single payer system.
But for expected care like dentist visits you can and absolutely should shop around.
Why? I'm not seeing any benefit to your idea vs single payer dental. It's not like your mouth isn't a part of your body or that dental issues don't effect your overall wellbeing.
Or you can sue them if you feel they should pay.
If someone can't afford insurance, what makes you think they can afford a lawyer?