The answer from Epicor Support to this question is no, it does not. Perhaps it needs to be part of the Edge Agent launch because I can’t run the auto-update from the smart client when the Edge Agent is running.
I did submit an Epicor Idea for this so if you have interest, please vote for it. Incorporate auto-update for the classic menu items using browser UX. There was another idea to Handle AutoUpdate with EdgeAgent running better so perhaps it is part of this idea.