We can view the category of $O_X$-modules as enriched over the category of sheaves on $X$, and I believe it should be the case that the sheaf of Sh(X)-natural endotransformation of the identity functor on $O_X$-modules is isomorphic to $O_X$ analogously. Not sure how helpful that is though.

This is a wonderfully clear answer, thank you very much. The particular section: "Perhaps we should think about this in light of the Stone–Weierstraß theorem: a 'good' notion of function should be able to separate points, and for this you need a notion of vanishing at a point." is what made it click for me - In order to have a notion of vanishing of functions we need to have an apartness relation (that respects the ring structure), and ring with apartness relation is the same thing as a local ring.

I want to avoid lengthy discussions on the nature of my question though - I'm seeking an explanation beyond "a certain amount of examples fit the theory, therefore the theory is good". That is a justifiable philosophy - theories are built around describing and abstracting certain examples after all, but it seems to me sometimes a deeper justification for particularly useful concepts can be found and that is what I was asking for.

@HeinrichD: Rings of germs of functions are also just plain simple rings. Yet the classifying topos of the theory of rings doesn't give me the same direct relation to lots of geometric theories as the Zariski topos does. We can consider Schemes as living inside the classifying topos of rings, but it will have the wrong colimits (glueing will give the "wrong" results). Also I didn't claim to "understand" schemes this way, just that definitions which seem otherwise ad-hoc follow naturally.

@HeinrichD: We could glue all sorts of algebraic objects together, say Monoids, but somehow mathematicians don't seem to think of Spaces equipped with a sheaf of Monoids as particularly geometric objects. (Someone correct me if that is wrong, one never knows how far people generalize things)

@HeinrichD: I was just trying to make clear that a lot of geometry follows directly from taking the theory of local rings and dissecting it naturally. In my opinion so much that "local ring" has to be some sort of fundamental concept in mathematics. I know of the "rings of germs" intuition, but this seems like an a posteriori justification. In some sense, yes - We think of a local ring $R$ as describing a local model structure and a geometric object being one that locally looks like what we've described. But what makes it that local rings are the right objects for this kind of description?