The Yale name will help in terms of networking and whatnot, but the market for legal academics has changed substantially in the last 5-10 years. In the past, the formula for academia was generally: 1) go to HYS, preferably YLS, 2) make law review, 3) get a big-name clerkship. The reason for this was that hires were based on judgments of prospective intellectual ability.
Now, law schools are looking to hire people that they see as having a distinct vision of their career interests and a rigorous sense of how their research agenda will progress. Going to Yale helps because they actively help their students shape this agenda (similarly, a Ph.D. will help in this regard for obvious reasons), but there is definitely also a sense that the Yale name will carry a bit as well. That said, I can't imagine that the Harvard or Stanford names will help you any more than Columbia, NYU, or Chicago would.
Ultimately, there are no 'prerequisites' anymore beyond publications and a coherent research agenda.
I would listen to this guy, he sounds knowledgeable. I've been hearing all of this as well. And going to HS over CLS full scholarship (especially S for those not looking to work in California) is just prestige whoring to the fullest. Really for someone looking to work on the east coast even without a full scholarship at CLS I wouldn't say going to SLS is necessarily better.