One of my friends last winter, walked back home super drunk and passed out on the street. Instead of being happy he didn't freeze to death, he was angry some people called the ambulance for him that took him to the hospital, and got a €600 bill for it.
I am not sure what's the common term for such place in English, Wikipedia is suggesting "Drunk tank": https://en.wikipedia.org/wiki/Drunk_tank
Police takes you there if you're too drunk to be on your own, it's paid (and pretty expensive).
No, it was for the ambulance ride. The socialized public insurance doesn't cover your drunk night out alcohol abuse.
And dental care is probably not free anywhere anymore that I am aware of, but its still cheaper to take trans-atlantic plane to get some more complex work done in cheaper places here than doing it in bigger cities in US.
[1] https://www.nhs.uk/nhs-services/dentists/who-is-entitled-to-...
*Emergency services are free in Austria but only for valid medical emergencies. If you voluntarily drink yourself to death like my friend did, they don't cover that and will send you the bill.
You're expected to have private accident insurance to cover such incidents and various sports and free time injuries that are also not fully covered by the mandatory government insurance.
Ambulances can be private. In Spain, if you contact the Social Security and it's them that send the ambulance, I understand it will be free. But if you call a private service, they will charge.
In a case like that, I would have called police instead. It's the safest option.
In countries where it's "free" one pays high monthly insurances, in countries with dysfunctional public healthcare systems services cost a fraction of what you'd pay in US but the salaries are proportionally lower.
Probably not so much in Southern Europe.