I have really quite enjoyed reading this thread, although I must confess I don't quite understand all of it. It has been many years since I have studied topology of any sort, and not only am I rusty, my mind is aging, and not as agile as it was in my youth. Anyway, for what it's worth, here is my take on basic definitions of topology. My first introduction to formal topology (rather than the specialized versions of it in real and complex analysis) was in simple homotopy theory, and the textbook I used gave the open sets axiomatically as a starting point. Although I saw (with some difficulty) that this was a generalization of the properties of the real vector spaces I was used to, and although it was in an algebraic style, I struggled with it's non-intuitive presentation. I believe (although I am hardly an expert) that "nearness" is a better approach to topology. It makes sense, even to a non-mathematician. For example, the statement of continuity in such a setting is simplicity itself: f:D->R is continuous at a point x in D, when: x is near a set A implies f(x) is near f(A). Furthermore, the axiomatic presentation of neighborhoods is simpler than the axioms for open sets in one important respect, you only need to consider the meet (for sets, intersection) of two neighborhoods, and the (possibly partial) order of the collection of the neighborhoods (for sets, the natural ordering on the power set, containment).
In a metric space, which automatically comes with a rich topological structure, you can easily resolve these notions to the traditional definitions. You lose the purely geometric flavor of topology in the process, though. Topology is concerned with matters in the small, and in the large, the nature of what are defined to be neighborhoods to a large part determines how intricate the spatial structure is.
As far as the aesthetic of why unions can be infinite, but intersections have to be finite, I have 2 thoughts: first, open and closed sets are somewhat dual notions, it is entirely possible to begin with the notion of a closed set, in which case you can allow only finite unions, but infinite intersections. In fact, this approach makes more sense for practical applications, since our physical tools for dealing with calculations (and our brains) are in fact finite. The second thought I have, is that when you take the union of two sets, you always get something "bigger", but when you take the intersection, you may get a null result. It seems natural to restrict the basic study to finite intersections, because infinite intersections can behave qualitatively different than infinte unions (similar to how, in whole numbers, substraction isn't always possible, but addition is).
Getting extra stuff is quite common in mathematics, you extend a field, or create a semi-direct product, or consider generated objects. But all this "extra stuff" is rather meaningless without some core thing that has some intrinsic behavior. In topology, I believe nearness should be that core thing. Much of topology's development was motivated by the idea of getting a handle on what a limit (and convergence) ought to mean, and these are notions which have their roots in approximation.
So, naively, one could argue, in real (one-dimensional) analysis we use open intervals as the basic building block, since we are often concerned about local behavior on very small intervals, and using that to extend to larger sets. extending this to a collection of open sets, or to a collection of neighborhoods can be shown to be equivalent constructions; however, using open sets focuses more on the "boundarylessness" of these sets (and thus emphasizes the density of the real numbers), whereas using neighborhoods focuses more on the "localness" of these intervals, lending itself more easily to abstraction while keeping some intuitive spatial idea.