The difference in the US is that it's (comparably) extremely difficult to change status from a temporary visa to a permanent one. Even if you
are highly qualified. For example the most common academic visa, the J1, is explicitly a temporary exchange program and you can't have immigrant intent (on application). Most universities won't give out academic H1Bs even though they're cap-free.
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.