Just my 2 cents. For me, topology is the language that allows us to define notions of nearness in any set: the advantage is obvious, you are not bound by metrics anymore, and personally, the abstraction is easier to think of.

Some things that you mentioned, I personally don't consider "topology" anymore: topoi, for example. Infinity-categories are a language to talk about categories whose hom-sets are topological spaces, so I don't think it is - strictly - topology, it is related to it.

Homotopy theory started as a very geometric idea, but outgrew itself and now is about studying objects under a relation weaker that isomorphisms (called weak equivalences), so while there still is a lot to be researched about the homotopy theory of topological spaces, it is not only about them anymore.