'That reminds me of an incident earlier in my career when a recruiter ranked candidates on their 'communication skills', which really was a proxy for whether they were an immigrant or not, in order to try to avoid discrimination laws."

There are a handful of successful immigrant CEO's in tech along with other industries (CEO of coca cola is just one example). One of the things they all have in common is excellent communication skills in english...so you can't necessarily write off that metric as an anti-immigrant filter.

This may be my opinion, but I don't know any tech companies that wouldn't hire someone just because they're from a different country other than for visa reasons.