Other than the unauthorized category (see e.g. https://www.pewresearch.org/short-reads/2025/08/21/key-findi... for estimates), it's pretty hard to get established legally in the USA too, and the criteria are similar to what you cite for Switzerland. By the numbers (https://www.congress.gov/crs-product/R42866#_Toc181884259), the majority of legal immigrants are family of people who are already citizens. The other big group is employment-related. Immigration for other reasons (specific diversity groups & asylees/refugees) is comparatively small.

American citizens generally do not think that they can just walk into another country and settle there, because you can't do that in the USA. A big part of what got Trump elected was that people just about everywhere, except the left wing of the political spectrum, were concerned about the scale of unauthorized immigration during the Biden administration. That does include a huge amount of asylum seekers under various programs, but even just CBP parolees at the southwest border totaled over 1.1 million July 2023 - July 2024 and that is a shitton of people (https://www.pewresearch.org/race-and-ethnicity/2025/08/21/u-...).

Also the USA has a literally 250 year tradition of people moving here (and an equally long tradition of fretting over how the "wrong" people are moving here). The entire history of the country is: people moved here and never stopped moving here, and along the way most of the native population were killed or exterminated by disease. You make it sound like some ill-founded 20th century liberalism run amok.

If anything, modern Europeans are too accustomed to people not migrating around. But it's worth remembering just how much migration (within and from without) had to happen before the modern European socio-ethnic layout emerged.