In fact, this is a poetic creation technique originated from The Book of Songs. The creative techniques of The Book of Songs mainly include "Fu", "Bi" and "Xing".
Fu is equivalent to the current rhetorical device of parallelism; Metaphor is equivalent to the current figurative rhetoric; Xing is to write something else first, and then associate it, which leads to what the poet wants to write most, or to express his thoughts and feelings, which is equivalent to the current symbolic technique.
For example, Guanluo, the first article in The Book of Songs, is associated with the pheasant in Guanluo Dove, which leads to "My Fair Lady, My Fair Gentleman". "Big saw, big saw" is what is used to arouse "interest", to generate associations, and to lead to the theme event of "Grandma singing big drama" behind.
What is used for fun has nothing to do with the theme? Of course not, but there is no direct connection between the two. For example, "Big Saw" and "Grandmother's Home Singing Drama", we can first see that "Grandmother's Home" is not in a bustling city, but in a relatively remote country, because only people in the countryside will be particularly interested in singing drama;
Secondly, "saw" and "play" rhyme. Perhaps, the first person who wrote this nursery rhyme couldn't find anything that rhymed with "play" at the moment, but someone "pulled a saw" in front of him and he sang it smoothly.
Summary: Incidental associations like this are very common in folk songs. For example, the lyrics at the beginning of the Xinjiang folk song "The Girl in Dabancheng" are "The stone road in Dabancheng is hard and flat, the watermelon is big and sweet, and the girl in Dabancheng has long braids and really beautiful eyes". Slate road and watermelon are exciting things in front, but they are different from Dabancheng girls' braids.