> I think the American First Amendment would obliterate this government body and probably the whole institution if it was ever tried.

This is not a government body, Ofcom is the relevant government body, like the US has the FCC, which you are aware of. The FCC has broadcasting rules. Your supreme court upheld their ability to issue sanctions for violations. This has lead to broad self-censorship by US broadcasters in much the same way the UK has Clearcast, to the point that censorship of stuff like swearwords is a recognizable trait of quite a few TV shows exported from the US. In the past year there have been multiple cases of censorship in response to threats from the FCC and other government bodies, much worse situations than banning an ad. The first amendment has done nothing to stop this.

I'm not here to defend the UK, they have some extremely scary laws on the books, but the US is really not notably different on this front.