The role of the Zariski topology in algebraic geometry

With the Zariski topology, generic points of varieties (which were known to algebraic geometers before the scheme formalism was introduced) correspond to actual topological generic points, i.e. those whose Zariski-closure is the entire space. Since the point of topologizing data is to introduce an appropriate notion of "nearness" to that data, and generic points are "logically near" every other point, this is a good indication that the formalism is useful.

I guess my answer would be that it's actually not that important that the Zariski topology is really a topology.

Topology is a pretty good axiomatic frame to describe the notion of closeness and continuity that some mathematicians came up with in the beginning of the last century. It's pretty good, that's granted. But it is not (even though some may have come to consider it to be) the ultimate formalization of these notions. It's just simple to formulate, quite easy to manipulate, and contains pretty much all examples that were needed at the time it was introduced. It gets the job done in an elegant fashion, and for that due credit must be given.

The Zariski topology on the spectrum of a ring is without doubt an intelligent and meaningful way to represent closeness of points in the setting, a notion that should exist given that we are doing geometry here (or at the very least we really want to think we do). And it is fortunate that it neatly folds into the dominant conceptual frame on that matter. It's really convenient and I think people back then were really happy with it.

But if it hadn't, well, I don't think it would have been the end of the world. For instance, when Grothendieck realized that there should be a finer notion of "covering" that captures more information, and that it was basically hopeless to try to define it as a classical topology, he came up with étale topology and the (essentially) more general notion of Grothendieck topology. It turned out to be a pretty good notion as well.

Basically, what I'm trying to say is that there are vague conceptual reasons that explain why it's not surprising that the Zariski topology is indeed a topology, but I don't think that too much importance and depth should be given to that fact.

Still, as an example of those reasons, topology was defined that way because people felt that it was natural that closed/open sets should have a certain lattice structure : as a convincing handwaving, a closed set should be somewhat like the set of zeros of some continuous functions, so it should be stable by intersection, and finite unions. Now a basic idea of algebraic geometry is that ideals of a ring can be thought as sets of equations, so they satisfy this special lattice structure for the same vague general reasons. Now you can see the construction of the spectrum of a ring as a special case of Stone duality : it only depends on this lattice structure, the prime ideals naturally arise from this structure, even though this is certainly not how they historically came up. So the Zariski topology is a topology because topology was defined following the same kind of intuition as algebraic topology.

(Actually, my reasoning is a bit anachronic, probably in a lot of ways since I'm not a specialist in math history, but at least because originally topology was formulated not in terms of a lattice structure on open/closed sets, but in terms of the closure operator. So the idea was that topology was a way to make sense of adherent points to a subset of a space. Of course all this is equivalent.)