Why are nets not used more in the teaching of point-set topology?

There are two places in the curriculum where point-set topology is taught. The first is a course in "general topology". Here the students have (hopefully) seen the basic topology of metric spaces (eg in Rudin's small book). Books intended for this audience (such as Munkres's book, which seems to be the gold standard) often omit nets and filters. I don't know of any written explanation from eg Munkres why he made this choice, but I can speculate. The typical student here is greatly inclined to think of topological concepts in terms of sequences. Teaching them notions of generalized convergence would be misleading. Given their lack of experience, they would probably think of eg nets as "just generalized sequences", not appreciate the subtleties of things like subnets, and in the end not appreciate the strange things that can happen in arbitrary topological spaces. Moreover, they would probably not learn to think of things like continuity in terms of open sets, which is much more elegant and conceptual and also quite important in applications (eg in algebraic geometry) where you are dealing with spaces that are very much not metric spaces.

The other place where point-set topology is taught is during functional analysis courses. Here certainly many standard books (like Reed-Simon) use things like nets, and this makes sense since the students are typically more mathematically sophisticated when they take these courses.

I've asked that question myself of both analysts and topologists, Calvin. There are essentially 2 reasons:

1) Firstly, believe it or not, outside of research analysts - who are really the primary experts and practitioners of point-set topology in modern times - many mathematicians either have forgotten or were never taught general convergence in topological spaces. Yes, it's hard to believe, but it's true in my experience. I used to know one of the officers in the Stanford University Student Society (someone correct me if I have the name wrong, please) and we were debating the usefulness of Riemann integrals when first presenting integration in calculus. I tried to argue in addition to it's intuitive mathematical value as a constructive limit, the Riemann conception gives a good example of a net. "A what? What's a net?" This is a guy who published an original paper on non-associative algebras when he was 20. My point is that at the top graduate programs, where the goal is mainly to race students to the research frontier as quickly as possible, most mathematicians just aren't being trained with these ideas since they're not considered essential.

2) Most mathematicians who are aware of notions of generalized convergence prefer the concept of a filter over that of a net. Filters are direct set theoretic constructions. As such, they are quite a bit more elegant and in some ways simpler to work with then nets, where the notation can get quite cumbersome. I personally agree with you, it's a vastly underused tool in mathematics. Most of the properties of sequences -which anyone who's finished a strong course in calculus will know well- generalize fairly directly to nets in topological spaces and that alone makes them worth considering.

By the way, the proof of Tychonoff's Theorem using nets is due to Paul R. Chernoff, it was published in 1992, I believe.

You should take a look at Albert Wilansky's book Topology for Analysis. In this book you will find the theory of filters and nets and you will also see how these concepts are related. It's a very good book. I prefer filters instead of nets, I think they are much more elegant, but as Wilansky says, one should not get attached. There are times when it's simpler to use nets and there are times when it's simpler to use filters. Just keep in mind that things that can be done using filters can be done using nets.

Some thoughts on why many authors might prefer the approach via opens:

See nets are nice if you want to grasp what being and getting close means in general topological spaces sure and this way in many cases give first hints at how to obtain a proof for some theorems. That is probably also the reason why you were able to reproduce a proof for Tychonoffs theorem mostly by your own: Intuition!

However there is a big conceptual subtlety about nets for describing topology: They provide in some sense an extrinsic description, namely how do nets behave in that space. Taking a closer look on this one encounters that unfortunately that happens in most approaches: Topology is described in most approaches by relating it to order theoretic structures, some of them include the approach via open or closed sets, via neighborhoods and via filters or nets. However most prominent the approach via open or closed sets or less prominent the one via neighborhoods are better in the extend that they choose a system of open or closed sets resp. neighborhoods while the approach via filters or nets compare them among each other.

Yet nets gain a lot of attraction when it comes to the point to construct specific objects in some topological space and I guess that is where one really benefits of them. Just to name some of them think of the Riemann integral, summability in general or more advanced the dynamics in quasi local algebras.

I hope that gave you some ease why still many authors rely on the old fashoined approach by opens.