The easiest way to relocate is usually to find a company that can sponsor your visa application, so that you're legally staying in the country, and can enjoy all the public services.
Health insurance a lees of a concern, in practice. Catastrophic coverage is pretty cheap from places like IMG Global, and doctors here are happy to take payment directly for more routine things if you’re not on the national plan. And that payment is forty dollars because there’s are no insurance companies to gouge, so gouging isn’t something doctors do.
Personally, I got around the “permission to live” thing by mildly abusing schengen tourist visas for a while, then marrying an English girl. If I had to do it again, I’d go for one of the “show us you have a ton of money in the bank” visa routes. Again, that US bill rate comes in handy...
I personally migrated by finding a company that would sponsor me, usually switching companies after that is pretty effortless.
A few years back I used to have an American friend who got here on a student visa and later changed to a regular one when she started working. Granted, she wasn't in tech or anything like it, but she always had a steady long term work contract. They never wanted to give her a long-term visa for some reason. It's one of the reasons she went back to the US.