The healthcare system is available from day 1 for people with working visas. There is also private insurance available which is 1/3 the cost of the US. Usually you get residency in a year, which also allows you to vote, one of only two countries in the world where non-citizens can vote.
But you are correct - getting in is not easy. You must have zero criminal convictions, great health with no pre-existing conditions, of a younger age, and a sought-after skill set. A small country can't afford to not be picky.