Composition is an old idea that predates Category Theory by a long shot.
But the real understanding of composability in all its complexity and generality has only come with (the more recent developments in) category theory.
Nope. It is just a different model for the same thing. You might find it enlightens your understanding of composition but it doesn't make it more "real" than any other model.
Historically, this mindset hasn't been productive, eg. complex analysts skeptical of GAGA faltered in the 90's and 00's. HoTT is still in its infancy.
What other model shares the same expressiveness and universality of category theory?