Doesn't Historical Linguistics suffice to train someone in etymology?
No, and let's work through the context of McWhorter's remarks to explain why.
Modern linguistics, since structuralism—in fact, since the neogrammarians—is about working out systems, and not just explaining particulars. It explains language as a system, through rules; that's what McWhorter is professing allegiance to. Etymology according to that mindset is a a throwback to the bad old days of the early 19th century, when linguists were not about the overall system but about the particulars; not about the forest but the trees. That's not how linguistics has worked for the past 150 years, and McWhorter finds no satisfaction in going through lots of minutiae about particular words, that are idiosyncratic and don't lead to a big systematic narrative. To him, it's sterile antiquarianism.
I can see why McWhorter reacts that way. I think it's a self-defeating reaction, because of course the majority of contemporary linguists regard the historical linguistics he practices as sterile antiquarianism. Historical Linguistics is far more popular with laypeople than with academic linguists, as I commented above. (Just as etymology is.)
So etymology's focus on the particular is not part of the modern linguistic narrative; and the modern linguistic narrative is overwhelmingly synchronic anyway. It is neither fashionable to teach etymology, nor ideologically sympatico.
But there's another reason why Historical Linguistics, even at a graduate level, is not enough to equip you for etymology. And it ties in to another question the same querent asked today: Isn't it obvious that linguists must specialize in language contact to study the history of English?
Language contact study requires deep knowledge of the history of both languages involved, to get all the details right: it is a cross-disciplinary enterprise. Etymology is even more cross-disciplinary: to get it right, you need to know not just several overlapping disciplines of linguistics (sociology, semantics, pragmatics, phonology, and their diachronic counterparts), but also several overlapping disciplines of history: cultural history as well as big-picture history (population movements etc). Just as philology, the other discipline modern linguists like to dismiss as Old Hat (when they're even aware of it) is cross-disciplinary, involving history and literature studies as well as linguistics.
No, contemporary linguists aren't trained in that kind of thing, and yes it is old-fashioned. But to dismiss it as a fetish and a tiny corner of your discipline is silly: it is in fact much bigger than your discipline. And those scholars who did not stay in the confines of linguistics or history, and wrote those dictionaries we all end up looking up, linguist or not, deserve huge respect. Especially because they don't get formal training in that line of work. (What cross-disciplinary scholar does now?)