Just because some (or even most) NPCs don't belong to a class, doesn't mean that class is only a PC "thing", that they are unique specimens, as was implied. First, that flies in the face of the term "class", which means that a member is of a certain type, i.e. not sui generis. Second, just because you use bandits or pilgrims in your game, doesn't mean that other NPCs don't belong to classes. That's kind of like saying that if I only throw beholders at my players, that proves classed NPCs don't exist. Third, even Ovinomancer agrees that "The rules clearly allow for, and have a strong assumption for, class in fiction" (whether this means they are structural to the world is kind of academic, because it depends on how you parse "structural" in this context), but the point is, the rules lean heavily in favor of the assumption that there are NPCs - many of them - who also have a class, and that they recognize that they belong to a class. The PHB suggests as much also: "While the fighter has contacts in a mercenary company or army, the cleric might now a number of priests, paladins, and devotees who share his faith". (p. 45). And then there is that thing about a calling, but that's already been discussed plenty in this thread.
As far as what Gygax intended - 1e had a required (not optional) system for level advancement and training, which presupposed coming into contact with members of your class to receive such training. For some classes, there was required alignment, combat with incumbents to advance, behavioral prescriptions, class languages, etc. All of this has already been discussed as well.
Last - the issue of classes being a convenience that provide a handy package for quick character generation: it has already been hashed out, several times, that this seems to suggest a skill system, rather than a class system, would be a better option for someone who thinks along these lines. I won't say that someone who thinks so is playing the game "wrong" - it's their game, and clearly, quite a few others think so, too. I will say, that if I thought along those lines, I would probably play a skill game. And moreover, if class was simply intended as a convenience, one might expect the rules to say so explicitly, refer to "classes" as "templates", and present a skill-based variant somewhere as an alternative (e.g. "for even greater customization, here is how to put together various elements, skills, feats, etc. to have a completely unique character").