You're right on the predicative complements.
Formally I suppose there's nothing wrong with calling the constructions be born and be going to die catenative. But I don't think you'll find many grammarians following that use. Be and be going to, like have and have to and be able to and the modals, are regarded as auxiliary verbs: what they contribute to a construction isn't lexical meaning but grammatical tense and aspect.
And there are restrictions on extracting an auxiliary from a chain. Consider, for instance, rewriting a catena as a pseudo-cleft. You can say:
He admitted stealing the picture → What he admitted was stealing the picture.
But you can't say
You've killed him! → ✲What you have is killed him.
I'm going to die a son of a bitch → ✲What I am is going to die a son of a bitch.