Not in terms of skills and proficiencies, it doesn't. When you pick up a new class, you don't get all the proficieincies, only some. Only a couple of them, if I recall, even give access to skills at all, much less the full sweep.
You are, of course, free to do what you like in your game. But, I don't think the rules as written are all that insensible.
What you say above makes some sense, when comparing Skills to the various artisan's tools. Becoming an expert on Religion might be seen as roughly equivalent to becoming an expert stonemason. I mean, the logic breaks down eventually - it takes more than 250 days to become a really expert in either, but there are simplifications to be made for the sake of playability and genre.
For the more adventuring-appropriate tools, though, the tools seem to be much narrower in scope than the skills. It seems reasonable to me that skills be harder to pick up than tools. The tools allow us a space to broaden character abilities somewhat, but keeps us from going hog wild on the matter, and making some of the choices at character creations somewhat more meaningful as a result.
Noting that the DMG preview lied in a subtle way, there may be a skill system variant or two in there that may give us something that we might both be happy with.