No, it's not dying at all. If anything, now is the best time to do homotopy theory. Thanks to the recent work of Lurie and others, homotopy theory is easier than ever to get into (advances have lowered the technical prerequisites) and has more power to touch an even wider array of other fields, via $\infty$-categories. Below are a few highlights from recent years.
2023: advances in algebraic $K$-theory allow for the disproof of the telescope conjecture.
2022: advances in algebraic $K$-theory allow for a proof of the redshift conjeture.
2017: Lurie's Higher Algebra, building on Higher Topos Theory, resolving the Cobordism Hypothesis.
2016: publication of work of Hill, Hopkins, Ravenel, using equivariant homotopy theory to resolve the Kervaire Invariant One problem (open since the 1960s).
2005-present: homotopy type theory
For more, see: