However, it's revealed in Skyrim that Thu'um is the language of the dragons. They say stuff like, I dunno, "fire", and their voices turn to flame. Nordic generals say "blow up the enemy's fortifications", and lo and behold, the enemy's fortifications get blown up. And, the player can wield this power for him/herself, presumably with the help of the Greybeards.
So, the Greybeards speak the dragon language, fine. But if they spoke normal Tamrielic at all, they wouldn't have to be all reclusive-like.
Maybe they can't speak Tamrielic? Maybe they don't have a choice but to never talk and stay away from humans in case they say something that the Thu'um takes out of context. Like, "my feet are killing me". Voila! Murderous feet. Obviously this is a gross oversimplification, but you get my point.
So, why wouldn't the Greybeards speak Tamrielic, unless they were completely incapable of it? Maybe it's their pesky lack of lips and all those razor-sharp teeth getting in the way.
So, yeah. They're totally dragons.