I very much agree with your observation that bind is not a very good choice when you want to explain the nature of the monad laws (i.e., the monoidal structure).  Instead (as you show), it's the Kleisli composition that makes it obvious.

I wish more people would use your line of explanation since it makes the monad laws look very natural.

BTW, I think strings are a good example of a monoid than everyone has encountered.

  -- Lennart