I can only speak for myself. The biggest Realms fan in our group has similar problems with the NPCs, though I won't go so far as to say that he will agree with everything that follows.
I have operated little in the Forgotten Realms, and it was over ten years ago, but the NPCs are indeed bywords for annoyingness. We played in Waterdeep, and the government of Waterdeep, to begin with, is whacked - the Masked Lord system is a paranoia-inducing one, perfect for secret cabals and corruption, yet the writers seem to think it's not subject to such flaws and could operate in a city cheek-by-jowl with an authority who's a paladin, and maintain authority over nobles and a rising bourgeousie, without generating significant conflict. Sorry, don't believe it. Also, sorry, neglecting a good source of adventuring for the intrigue-inclined. What terror of Undermountain can compare with the growing conviction that some of the Masked Lords have it in for you, and moreover pose a danger to your beloved city? A conviction you cannot prove...
A lot of the NPCs, when taken as written, don't act as real people would - their motives are obscure, contradictory, or laughably unrealistic, and the people explicitly commended by the authorial voice (whom few assume might be unreliable, but who I think is best read that way) are often acting either against their own best interest or in an inappropriate way for the setting.
Khelbhen specifically is annoying because he's simultaneously so powerful, so bossy, and so secretive. If he asks you to go somewhere and do something, you're not supposed to ask questions, and won't get good answers if you do, if he's played as written. You need dental implements to get information out of him - magical ones, at that.
Elminster's the one I really hate, though, because Greenwood uses him as a mouthpiece a lot, and, I'm sorry, Greenwood overwrites, with the result that Elminster is a blowhard of the first order. Read any speech of Elminster's at random. If this guy were sitting next to you on the bus, wouldn't you have to get off at the next stop to restrain yourself from slapping him as he droned on and on and on and on and on and on...His conveys useful information, sure, but sorting out the good information from the character bits is a full-time occupation. There's an art to doing infodump and characterization at the same time. Elminster is a good example of how not to do it.
Volo of the Volo's Guides has a similar problem. In fact, I have a hard time telling his voice from Elminster's. I won't go so far as to say that everybody in the Realms speaks with the same voice, but I will say that there's way too much overlap, and that attempts to create a distinctive voice often result in either an improbable and difficult-to-read accent, or an even more tedious variation of the baseline authorial voice. Greenwood does not do an adequate job of writing disparate individual voices, and authors who aren't Greenwood feel obliged to sound like him.
What everybody I've heard discuss the question agree, however, is that the major NPCs are just too powerful, and it's hard for the PCs to feel that there's any point in anything they do. If they fail, the gods can descend in their machine to fix it, and in fact, if they weren't going to give the PCs the information they needed to do a good job to begin with, they probably should have. Consequently, a DM has to either take great care, or go to great lengths, to create a challenge which doesn't seem trivial. Ours did it by picking on our characters as individuals. At one point, when my character was trying to become pregnant, I and my in-game husband cornered him and told him, in no uncertain terms, "No demon-babies!" 'Cause he would've done it, just to create a personal problem that we were the obvious people to deal with.
I think the strength of the Realms is its detail - all those maps, all those names, all the existing politics and trade routes and gossip and the sense that you can move in any direction inside an existing world. It has depth and breadth. But the characterization of the NPCs and the extravagance of some of that detail (just as in college towns you may need a degree to flip burgers, we joke that you need class levels to wait tables in Waterdeep) are a big weakness, and should be taken in hand by the DM early in the campaign process.
What happens to the Realms if you take the big guns out of it? What if you throw out the bits of character write-up that annoy you or make no sense? What if you take the bits that annoy you and say: "Okay, this NPC is annoying in this particular way, and instead of fawning all over him, the other NPCs treat him as a bore, or an obstacle, or an object or ridicule, or someone to fear rather than love, depending on the nature of that character flaw."?
