I am an undergraduate with huge interest to the topology and foundations of mathematics. I have been studying Spanier's Algebraic Topology and Engelking's General Topology, and I noticed that while algebraic topology heavily depends on the category theory, set-theoretic topology does not.
Is there a reason why category theory is not used in set-theoretic topology? Is there no advantage of trying to introduce the categories to the set-theoretic foundations? At least from books in general topology, I did not see single use of categories except for spaces relevant to algebraic topology like quotient space.
There are advantages in the use of category theory in the presentation of general topology, or any general theory.
One aim of category theory is to look at an object or construction in the light of its relation to all other objects through the notion of morphism - for topology the morphisms are the continuous functions. In this view the definition of a topological space (by open sets, closed sets, neighbourhoods, ...) is less important than the morphisms, and the global properties of the category of topological spaces and continuous maps.
This consideration led to the notion of convenient category of spaces, see also the Introduction to this 1963 paper.
Further defining constructions on, or properties of, spaces by their categorical properties allows for analogy and comparison with other categories. As one simple example, in my book Topology and Groupoids (T&G) I define a space $X$ to be connected if the only maps from $X$ to the discrete space with two elements are constant. This is sometimes useful for proofs of connectivity.
Here is another example: the Tychonoff topology on the product set $X= \Pi_{i \in I} X_i$ of a family of topological spaces $X_i$. This topology may be defined purely in terms of the open sets of the spaces $X_i$, and this is useful to know. However there are projections $p_i: X \to X_i$ and a key property of these which we want is that a function $f: Y \to X$ from a space $Y$ is continuous if and only if all the projections $p_i f: Y \to X_i$ are continuous. This property characterises the Tychonoff topology, and gives an analogy between this topological construction and properties of products in other categories.
Such more categorical presentation, and so further possibilities for analogy and comparison, gives prospects for analysis and evaluation, whose necessity for progress in science is discussed in this article by Einstein.
The two books you mention were published quite a while ago, when the wide value of category theory was not so clear as today. Also, some people are intrinsically interested in problems, others interested in the development of structures and viewpoints. Each should be encouraged, as a part of the methodology of research.