This isn't obvious at all. A few very specific, niche employment areas in which the US dominates (software engineering in certain niches, medical practice, and maybe show business) have a modest influx of Western European workers, but that says nothing.

Among the reasons for emigration to the US, there's not a single living soul that says "oh, I'm not legally allowed to work enough in my area of employment to achieve my dreams". It's a laughable and ridiculous statement.