Even if the books don't explicitly tell you ''look, if you wait for too long, this [instert baddy here] and their lackeys just *might* do some pretty heavy damage'', it should be a plausible conclusion to draw. Just try to imagine what a PC (and I mean in-world character, not the player) would feel when knowing that their world is in danger. It is just natural that they would feel pressed to act quickly, like they were racing against time, because their lives and their world are in danger (and because they know that the world doesn't revolve around them, nor the uber evil will wait for them to get strong enough).
That said, a campaign where the world revolves around the PCs and nothing major happens without them (unless it is already written in the plot) is as valid as any other gaming-style, but it's not everyone's cup of tea. With that, you would solve the problem of hyper fast, prodigy-level learning, but it would be a trade off for uber evil, world ending threats not appearing as really world-ending and not giving a feeling of serious danger. That's why (IMHO) high-stakes storylines shouldn't start at low levels, or, if they did, they shouldn't supposedly cover a short period of time.