j'éviterai ainsi de parler aux pestes - presumeably this has a meaning other than "I will thus avoid talking to the pests"?

Martin, 

I think it is a reasonable guess that this is figurative  ‘ aux personnes méchantes ‘ ou ‘ aux casse-pieds ‘ - just as ‘pest’ in English has similar figurative interpretations :

https://www.wordreference.com/fren/peste 

 https://www.wordreference.com/enfr/pest 

On the other hand, it seems to be a fun, creative rewrite by the class of a children’s ‘educational’ song, so anything is possible I guess!

My query was really about how being "modeste" links to avoiding talking to pests?

Martin, 

It rhymes - pretty sure there is not any other deep reason !