I have been doing what I consider to be higher category theory since the 1970s. It went under the name of homotopy coherence theory and so was thought of as homotopy theory. I used Kan complexes etc.Which is it? The distinction is TOTALLY ARTIFICIAL so, Chris, I do rather object to setting up strange barriers between different areas of pure mathematics as this question requires.
That was more a comment, here is an answer:
Ronnie Brown and Phil Higgins' work on higher dimensional groupoids is a natural continuation of J. H. C. Whitehead's Combinatorial Homotopy II. The first of the two papers is where CW-complexes are introduced, so is that algebraic topology. Could RB and PJH have proved their higher van Kampen theorem without using higher dimensional groupoids? Possibly but they didnot!!! They used intuitions and methods from higher dimensional category theory .... including ideas from Grothendieck, Ehresmann, Benabou, Kelly, Street, etc. (The person (not you, Chris) who asked the question is blinkered if they think that mathematics divides up neatly into bits of independent subject areas with no interaction.)