It would be that the 'powers that be' don't allow mortals to wield power associated or derived from them.
It
would, however, mean that there would be no profession in the world dedicated to the expression of said beliefs, as the mechanic of 'classes' reflects a setting's common professions in D&D. While such beliefs may
ostensibly exist in a setting, if they don't benefit from the standardized D&D mechanical representation for such things, is that existence of any consequence? I would argue that the only possible answer to this question is "no"
[Note: Incidentally, this is what I was referring to when I said "I think that such beliefs and
the representation of believers in the world is a vital component of compelling fantasy, for me. Without the classes, the representation of believers is notably lacking.]
As with the human-centric bit I mention earlier, it's one thing for a product to say "X things exist this way in the world!" and another thing to support that claim. D&D has a long history of claiming X and then uitterly faling to provide mechanics for it (or, in the case of claimed demi-human obscurity, actually printing a metric ton of material to the contrary). This has, similarly, been one of my long-standing complaints with D&D setting design.
Saying that belief systems or believers exist in a setting but completely failing to apportion them any kind of mechanical representation isn't much different than omitting such things from a setting entirely, in my experience (YMMV, of course).