But do you see something equivalent withheld from your paycheck in the US? Not that I noticed, when I was working there.

I don't know, I've never worked in the US. OP seems to imply you do?

I have before. I'm confused why people are bringing up employer NI if there isn't some kind of similar deduction that comes directly out of employee paychecks in the US. Otherwise, who cares that your employer pays some tax for everyone that they employ?