Personally, in my FR games, I greatly, greatly tone down the NPC badasses in the world. I don't follow the canon so it's easy to do.
I'm not sure how many NPC badasses there are in the Realms at present, actually. Drizzt is a great fighter, but nowhere near the power needed to take on a demon prince singlehandedly. Elminster may be less powerful than he was (and has his own problems), and after that the Realms are really in a different state from during the 3E era.
There's also the problem of these NPCs actually leaving their homes: it leaves them undefended, and they have enemies that are happy to take advantage of it!
I'm currently reading through the Companions Codex trilogy, set in the period between the Sundering and Tyranny of Dragons. An orc horde (sponsored by the drow) attacks the Silver Marches: Nesme falls, and other places are besieged for *months*. This is in the same area of the world as Out of the Abyss, and no Elminster is coming to aid them.
Who's the best wizard nearby? Probably Laeral Silverhand in Waterdeep, Open Lord of the City. Is she going to leave Waterdeep undefended to travel into the Underdark on mere rumours of trouble there? (Menzoberranzan has been a major source of worry for the surface people for centuries, but it's not being attacked...) No, she's going to send agents to find out what's really going on. That'd be the PCs.
The question is what happens next, when the threat is identified. In
The Rise of Tiamat, cities all over the Sword Coast sent armies to confront the Cult of the Dragon (and were kept busy throughout with the destruction caused by the cult). It's one of the reasons that
The Rise of Tiamat is one my favourite adventures of all time: it really displays the world working in response to a threat, rather than just letting the PCs handle it. Sure, the PCs are the major agents working against the cult, but that's fine.
I still haven't really pinned down Princes of the Apocalypse, despite having now run it for many sessions. I don't really get the same sense of the Elemental Cults proceeding towards a goal. What they're doing affects the valley and little else - so it isn't that surprising that the other powers in the region aren't involved. (Once they do find out, well, they probably feel the PCs can handle it...)
There's a lot of unanswered questions about the structure of
Out of the Abyss, so it's impossible to properly discuss it at present.
Cheers!