Whoops, was working from memory and it has been a looong time. DH key exchange is indeed completely different to what I said above - as you've said it is an alternative way to set up a secret channel. Certifying trust on a public key is indeed the important issue, and solved in a different way.
I'm guessing the alternative issues on IBE are the need to trust the CA, which puts it roughly in the same level of messiness as using digital signatures on public keys anyway. Is that Matt Palmer at the National Archive?