I am not going to enumerate all the reasons why it’s a nonstarter, but at least one is that it massively increases the cost of any such social program to a point it becomes both practically and politically untenable.
I’ve visited and used healthcare services in many of those other Western countries as a foreigner, and every single time I was required to file paperwork and pay a deposit in cash (or rarely by card) before receiving services. I would not have expected any differently, as I’m not a tax-paying citizen in those countries. On a mostly unrelated note, I write this while currently residing in San Antonio (as used in your example scenario).