The leaders/parties supporting immigration in those countries are ambivalent to receiving high skill immigrants or refugees.
The US, UK, and Canada all have special provisions in their immigration programs aimed at attracting and prioritizing highly-skilled workers.
Both the UK [1] and Canada [2] both use a points-based ranking system that prioritizes highly-skilled immigrants. The UK system is clear in its goals:
> introduce an Immigration Bill to bring in a firm and fair points-based system that will attract the high-skilled workers we need to contribute to our economy, our communities and our public services.
And while the US H1-B program is lottery-based, 20,000 slots are reserved for people who hold a master's degree from a U.S. institution. Proposals have also been made recently to change to a points-based system. [3]
> They built themselves into places that high skill immigrants seek, but that is more of a side effect than a competition
Wherever there is choice, there is competition. 55% of billion dollar startups in the US have immigrant founders, employing an average of 1,200 employees each [4]. If these people don't come to the US and start companies, the US will feel the effects - even if they were just "side effects".
[1] https://www.gov.uk/government/publications/the-uks-points-ba...
[2] https://www.canada.ca/en/immigration-refugees-citizenship/se...
[3] https://en.wikipedia.org/wiki/RAISE_Act
[4] https://www.fosterglobal.com/blog/55-of-americas-billion-dol...
This choosiness is actually a sign that immigrants are competing to enter those countries. The points based system is (in theory) a way to identify the ones we want.
That said, illegals and “refugees” outnumber H1Bs, further reinforcing that Western countries don’t care about global talent.
It goes both ways. A more streamlined application process and straightforward path to permanent residency is a draw to would-be immigrants who qualify.
I won't discuss illegal immigration or asylum here as those exist for different reasons, other than to say that it's a logical fallacy to assume that just because A is bigger than B, a country doesn’t care about B.