The Yeshivish world definitely learns kabbalah — it just depends on how you define kabbalah. Zohar? Perhaps not. But the idea that "Zohar=kabbalah" is quite misconceived. Indeed, I know that quite a lot of the Yeshivish world learns the Nefesh HaChaim, which is quite a kabbalistic sefer. While they may not go too deep into the more esoteric ideas — and certainly do use it as a mussar platform — even just reading the basic text is quite kabbalistic and mystical. There are other books that are rather kabbalistic in nature that are widely read by the Yeshivish world like Derech Hashem, some of the Kuzari, as well as much of the Ramban's peirush on the Chumash, and so on.
As to why many people tend to stay away from the Zohar? There are a number of reasons, not all of which is this the proper place for. But in short, it is deeply esoteric, very difficult to learn, hard to understand (properly, or even at all), and is known to have some sections that are forgeries. Thus, if you're going to learn the Zohar at all, it is much wiser, unless you are a true scholar (and I mean that in the greatest sense of those words) to do it through, say, the Vilna Gaon, or the Arizal (who both wrote commentaries on the Zohar).