Agree with your thoughts. I always enjoy this amusing poem to congradulate Frank on the birth of his second child, a son. It is funny, very touching and shows so much family feeling by Jane.
Jane also expresses happiness on the move to their Chawton home.

I think families with literary taste like the Austens spent much of their leisure composing verse for amusement to be read aloud ,and, sometimes to commerate events in the family.
Interestingly, I cannot recall Jane writing of any families or a charecter who composes verse in her novels.