[Manuela]

Hi Mark,
I don't see any use in teaching students such chants either.
I do the same groupings you do. And I teach him, her, them in a sort of by the way way, while teaching other things. For example:
(after inviting a student in front of the class.
Look at him! What does he look like? (He is tall, with short, brown hair and blue eyes). or Look at him. What is he wearing today? or Look at them. What are they doing? (when students have been asked to mime doing something.) Look at him! What's his name?
Look at her! What's her name! etc.
or whatever else I am teaching. So they simply get to use those pronouns without me saying that those are pronouns.
I really don't think that chants help you to actually use the words you remember with their help.
Such mnemonics may come in handful for let's say remembering the order of adjectives before a noun, for example. I remember this OSASHCNM (for opinion, size, age, shape, colour, nationality, material) but of course this kind of thing is only useful when taking exams when you have to tick the correct answer. So basically even this is useless when you actually speak. It may help you when writing a composition to check things out before handing it in, maybe.
Well the Beatles were sort of remembering this chant too, weren't they? Didn't they have a song entitled I my me mine ? or something like that?
Well, happy teaching
manuela