In the same way that "I" isn't part of the first line of "I am the very model of a modern Major-General"? That "I've" isn't part of the second line? That the first "and" isn't part of the line "And hydrogen and oxygen and nitrogen and rhenium", even though the other three are? That only the second half of "in short" belongs to the line "in short, in matters vegetable, animal, and mineral"? All of these are in metrically identical positions.
You really think a poem in which every line begins with an identical anacrusis, where the "anacrusis" is required without exception, is properly analyzed as using anacruses?
You might try listening to the song before making an obviously false claim.
Here's an official PDF with lyrics: https://tomlehrersongs.com/wp-content/uploads/2019/02/the-el...