*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.
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-...
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.
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.