Higher category theory is, roughly speaking, where category theory meets homotopy coherent mathematics. It is hence relevant to those problems in which categorical structures and homotopy coherent phenomena play a significant role. Many areas of algebraic topology and algebraic geometry have this property. There are also many such areas who don't. From what I understood from your question, you like category theory, but not so much homotopy coherent mathematics. So far I would say you don't actually have any problem, since ordinary category theory itself is not, at least in my opinion, a domain in which homotopy coherent mathematics is crucially needed. This is mainly because the coherence issues that arise in category theory are very low-dimensional, to the extent that it is more cost effective to do them by hand (or simply neglect them), then to use fancy machinery. This leads me to the first possible solution to your problem:
Do category theory.
It has really not been my impression that this field is anywhere close to finished. This is especially true if you consider 2-category theory as an acceptable extension (here the coherence is again usually simple enough to do by hand). It also has many interactions with domains such as logic, set theory and foundations of mathematics. You will find many interesting discussions of all these topics, as well as links to state-of-the-art research, in the n-category café. It will also not surprise me if you will find people there who share your mathematical taste.
If you still maintain, for whatever reason, that it is imperative that you do things related to higher category theory, I can tell you that there are many domains in this topic which are very 1-categorical in flavor. For example, you can
Do model category theory.
This idea, one of many brilliant ideas of Quillen, allows one to magically reduce homotopy coherence issues into a 1-categorical framework. Model categories also share many of the aesthetic features of ordinary category theory, in the sense that everything seems to fit together very nicely, while still being extremely useful for real world homotopy coherent mathematics. A bit less known, but also very categorical in flavor are derivators. You might also look into triangulated categories.
Finally, as many of the comments above suggest, it's possible that the things that you don't like in homotopy coherent mathematics are actually not essential properties of the field, but rather of its young age. You may hence consider
Give HTT another try.
In doing so, you may want to take into account the following: I strongly believe that no one has ever written a technical simplex-by-simplex combinatorial proof of an HTT-type result without knowing in advance that what they want to prove is true, and moreover why it is true. This is because, despite the technicality of some proofs, higher categories do behave according to fundamental principles. Sometimes these principles are the same as the 1-categorical case, but sometimes they're different. As a result, it may take a bit of time to acquire a guiding intuition for what should be true and when. It is, nonetheless, certainly doable. In doing so, I would suggest that, before reading a given proof, you try to think why the announced result should be true. In addition, think how you would prove, say, the 1-categorical case, and then try to extend the proof higher dimension by dimension, and see where it leads. Then read the simplex-by-simplex argument. It may suddenly look very clear.