I, like you, started with a Basic D&D years ago, and progressed. But that experience (no doubt romanticized by distance by both of us) is not really relevant to the discussion of 5e.
KS
It's absolutely relevant, as it appears to be the model 5e is following. To me, it seems obviously they've set the game up to create an easy Basic boxed set of the game, with no skills/backgrounds or feats and just one class from each of the four main class groups, all in a fairly thin set of rulebooks (or maybe just one book) along with dice and some sort of sample adventure. Other...comments from people being paid to consult for 5e but who are not on the design team itself...suggest (without outright saying it) this is indeed part of the plan.
So if the plan (as it seems to be) is to create a Basic boxed set of the game to sell in Toys R Us, Walmart, Target, K-Mart, toy stores, bookstores, and other stores like those, then past results from such a similar boxed set is relevant, and things that would assist in transitioning from such a Basic set to a more Advanced game are relevant.
Now imagine you learn the new game from a Boxed set you purchased as an impulse buy at Target, having heard about D&D from an Ad or news over the years or a Facebook post or whatever. You play it, and like it, and want more of that game and see the ad in the back for the Advanced edition. The Basic edition includes four classes, a Fighter [Warrior], a Cleric [Priest], a Thief [Rogue], and a Wizard [Mage]. You look at the Advanced edition and you see things like Barbarian [Warrior], Druid [Priest], Assassin [Rogue], and Sorcerer [Mage], along with a bunch of other classes that you don't recognize but they all have those tags after them that you do recognize from the Basic game.
That sense of familiarity, that the Advanced game is like the Basic game just with additional similar stuff, helps you make the decision to buy that Advanced game. It reduces the intimidation factor inherent in buying such a big book as opposed to the little book you already bought. It suggests the stuff you read in the bigger book will be similar enough to the stuff you already know that you won't be bogged down by thousands of new rules and details that are unfamiliar.
That, to me, is a huge advantage of using these classification tags. It might not help you transition, heck you probably won't ever play the Basic game. But it will help someone new to the hobby transition from the Basic to the Advanced books. And it will help again when they look at the supplements that have those tags as well. That makes it a useful tool for the game in general, though not necessarily helpful for you and your group.