Maybe I'm missing something here or I'm just being dense but what exactly is the problem with EtM and AAD? Just use a suitable encoding like MAC(len(aad) || aad || E(plain))
Well, when that happens, I feel the need to pipe in :)
If you're careful enough to not implement a naive protocol that stitches AES+CBC and HMAC-SHA2 together (or, as tptacek put it in a podcast episode, throw some crypto potions into a cauldron and see what happens), you're probably the minority of crypto-savvy people.
They feed an IV and ciphertext into HMAC. They don't feed additional authenticated data.
If someone followed Signal's example, they either wouldn't have AEAD, or they're likely to make the exact mistake described in the post I linked above.
I don't know how to be more helpful here. I've been only repeating myself.