It's definitely coherent, and I agree that plenty of fanfics lack a decent representation of some of Grantaire's other qualities, but I definitely don't think it's fanon to assume he has low self-esteem or is depressed. He describes himself as melancholic and displays strong mood swings and bouts of melancholy. Fanon definitely tends to woobify him a bit, but Grantaire shows symptoms of depression from a purely clinical standpoint.
And he is drunk in many of his scenes, is known for his drinking habits, so it's not entirely fanon to assume he does drink quite a bit. He also has a very distinct way of cutting himself down in his speeches that makes it seem like his self-esteem is not entirely intact. He puts on a good show, but that's the entire point of his character. Everything about Grantaire is a show. He doesn't just say he wants a drink - no, he wants to be a leech on the tun of the Heidelberg. He doesn't just sit in his chair; he sits with his fist on his knee, arm bent into a right angle. He doesn't just laugh, he bellows from a window and harrasses the serving girls.
Grantaire tries very hard to be a lot of things - a non-believer included. I would say he's far more skeptical than cynical. I think it's very important to note that he's not described as not believing in anything, but in being a man who takes great care not to believe in anything. There's a big difference between those two descriptions. In the one, he might be incapable of belief, or have simply decided he doesn't care, or for whatever reason have simply stopped believing in anything. In the latter, he puts effort into not believing in things, which begs the question why and also means that he might not always succeed in completely not believing. He won't let himself believe, but a part of him clearly wants to, considering the way he's attached himself to Enjolras. And I want to write more, but I'm exhausted and can barely put my thoughts into something resembling coherency, so maybe tomorrow!
I mean, he's clearly much more, and I'd love to see more of his encyclopaedic knowledge of Where-to-go-for-what, his gambling, his boxing, his flirting and everything else in fics, and I agree the depressed alcoholic with a bad childhood spiel can be easily abused and cut his character down to just that, which is vastly disappointing, because he's so complex and fascinating; but I don't think that the basis for those themes is un-canon or incorrect at all.
"He's a faithful and loyal friend, he isn't interested in working or change, and he often thinks about the world without taking an active role in it." <-- Definitely agree with this, though.
I hope this made sense. My thoughts are a complete jumble right now.