Do all nursery rhymes satirise public figures or events?
Probably not, but many do. Many more than we realize. Have a look into a casual little book called “Words Lightly Thrown” which explains the origins of many nursery rhymes. For instance, in “Little Jack Horner” he sticks in his thumb and pulls out a plumb … this apparently relates to when Henry VIII confiscated a lot of church lands – the deeds and documents would need to be transported. It was common practice to bake these into pies so that they would be hidden from highwaymen and thus transported safely. Jack Horner (and his real name escapes me, but they detail it in the book) reached in and plucked out one of the documents and secured ownership of the estate in this way and according to the book, his descendants still own the land to this day.
