Those words haven't (so far as I know) been sung either: they're from one of his spoken preambles, not from any of the songs.

(The preambles are generally at least as funny as the songs, at least in my estimation.)

From https://tomlehrersongs.com/we-will-all-go-together-when-we-g...

I wouldn't call what he does singing given the normal meaning of the phrase.

That seems unfair. He is (or at least was when he was doing it -- he's 96 years old now) a very competent singer and pianist. He's not particularly trying to make a beautiful sound, but there's more to singing than that.

Even he has said that calling what he was doing 'singing' would be generous.

Well, yes, self-deprecating humour is a thing, but there's a difference between Tom Lehrer pretending he can't really sing and someone else claiming he can't really sing.

mini-musical?