Cardinal arithmetics is usually considered far removed from homotopy theory,
so perhaps this would be an example.
Several cardinal invariants in set theory may be viewed as derived functors, in a very degenerate model category setting. is usually considered remote from homotopy theory. See Gavrilovich, Hasson, A homotopy theory for set theory, Part I, Part II.
In fact, there is also the following which appears to be an explicit attempt to
that the ideas of homotopy theory have very broad applicability in mathematics.
Gromov, In search for a structure. Part 1, On Entropy", talks entropy in terms of category theory, but perhaps not necessarily homotopy theory. He has the following to say in a postscriptum:
Apology to the Reader. Originally, Part 1 of ”Structures” was planned as about a half of an introduction to the main body of the text of my talk at the European Congress of Mathematics in Krak´w with the sole purpose to motivate o what would follow on ”mathematics in biology”. But it took me several months, instead of expected few days, to express apparently well understood simple things in an appropriately simple manner. Yet, I hope that I managed to convey the message: the mathematical language developed by the end of the 20th century by far exceeds in its expressive power anything, even imaginable, say, before 1960. Any meaningful idea coming from science can be fully developed in this language. Well..., actually, I planned to give examples where a new language was needed, and to suggest some possibilities. It would take me, I naively believed, a couple of months but the experience with writing this ”introduction” suggested a time coeﬃcient of order 30. I decided to postpone.
There was also talks by Voevodsky (and older work by Tsvetkov(?)) who consider a category theory approach to probability. (There is an online
video talk by Voevodsky on this in Russian. I'll try to recall the name of the book by Tsvetkon(??))