For "gringo" all that's needed is a nasalization of the central vowel of "griego." (It happens in languages all the time: French for example. It's roughly the reverse of "dropping g's" in English, and we know how easy that is.)

It's harder to go from "green grow" to "gringo." Try it. Furthermore, every Spanish-speaking Mexican was familiar with "griego." No more than a handful could have been interested in "green grow" even if they'd heard it - and it's only an assumption that anyone did.

And of that handful, no more than *one* could have thought, "Wow! I'm gonna call all North Americans 'green grows,' because I've heard some of 'em sing that song! And I won't pronounce the second 'r' either! It's sooooo cool!" Then his friends would have had to think it really was cool; otherwise they wouldn't have started using it themselves.

As with everything else in the world, one cay always say "what if?" and "maybe...," and "that's just your opinion." But "griego" is the only choice that's supported by both fact and probability.

(It would still be even if the soldiers did have a song that started out with "Green grow...")