As someone who is a US Citizen, what is something about the immigration process that I probably do not know about, but causes a lot of issues/could be improved.
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Edit: I am taking a break now and will return later this afternoon/evening to respond to any comments and answer any questions. Thank you everyone for a great and engaged AMA so far.
No other country in the world requires foreigners to be significantly more qualified than its own population. You can move to France with a regular paying job no problem or just a few thousand euros in savings. Impossible in the US. You have to be extraordinary (they literally call their criteria, "extraordinary abilities") or you have to make top 5% money (so if you work in tech, that would be at least $500k-1M/year in many cases).
The only other way is to get married. This means there is a massive discrepancy between the qualifications of self made immigrants, versus those simply lucky enough to fall in love. It's pretty unfair, but that's how it works. But that's also the reason so many immigrants are so successful in the US, the bar is so high, that it creates a massive motivation to succeed to become eligible for the criteria.
Unless something has changed dramatically in the last decade this is patently false. Getting an EU work permit was historically very hard with employers having to demonstrate that a position can't be filled by an EU citizen before a non-EU citizen candidate can be considered.
In most European countries, once you're in, you can find a way to stay. One exception I can think of is Switzerland, which can be pretty annoying for temporary visas because they don't count for time accrual.
Austria has a pretty good system (RWR) that lets you job seek and is a pathway to permanent residency as a 3rd country citizen. I think there are similar programs in France and Germany.
For example "very highly qualified" in Austria is satisfied by almost anyone with a STEM degree, being under 35 and (amazingly) being an English speaker. If you have that initial visa, companies can hire you without worrying about sponsorship.
You could also use that as a route to the Blue card I think. I wouldn't say the bar is exactly low, but a lot of mobile people are sufficiently educated and are paid enough. As in, a typical European STEM salary would cover it.
But also the grandparent's comment is out of touch. Of course countries want people who are more skilled than local labor, that's the whole point. Aside from the benefit of attracting talent and higher tax revenue, it's much harder for your voters to argue that immigrants are taking your jobs this way.