How often do you see Europeans work remotely for US companies? Is it extremely rare in your experience?
In my experience yes but I live in small country in western Europe that's never on the radar of recruiters from the US, unless they're looking for a holiday destination :).
I never found US tech jobs open for remote candidates that were not some off-shore sweatshop kind of slave work looking to undercut the local US market wages with candidates willing to work longer hours for less pay than the local talent.
The good US tech companies never put their crown jewels in the hands on remote workers from abroad they haven't already known before or gotten through some internal referral.
I'm sure there are a few exceptions and glad to be proven wrong though but that's what the market looks like from my PoV.