Most countries with free healthcare systems like you describe don't officially give that benefit to visitors from the US or other countries, with certain common exceptions like travel within the EU/EEA and people with some form of long-term residence permission like a work permit.

They will of course allow anyone to access the system. And often they don't charge because they don't want to deal with the paperwork. But they're even more often officially supposed to charge.

My (American) father once needed emergency healthcare in France. He eventually received a bill from the French government in the mail back in the US. It was way cheaper than in the US, certainly, but it was not a zero balance due either. (And yes he paid.)