Is homotopy a consequence of making paths composable and invertible?

Question

Many introductions to algebraic topology go like this:

1. We want to study spaces by studying paths in them.
2. It is intuitive that one wants to compose paths, but the naive way to compose them fails horribly.
3. this is fixed by considering paths up to homotopy
4. the notion of homotopy can be defined for arbitrary maps, not only paths, so lets do homotopy theory…

From a applied perspective this is of cause perfectly fine. But it was (and still is) unclear to me exactly why we use homotopies and not some other tool to fix the composability issue of paths.

There are some obvious „fixes“, which are sometimes discussed:

• one could try to simply use the images of the paths. Thinking of loops this is of cause not a great idea, because we cannot distinguish the number of times the loop winds around a hole. So we drop this approach.
• another idea is to use intervals of different lengths. This gives a strict category of paths in a space $X$, but it is not a groupoid. We would like paths to be invertible, so we have to fix this somehow. This leads us to reparametrizations, but then the whole „different lengths“ business isn’t really necessary anymore.

So we agree that we somehow want to define a groupoid of paths in $X$ and we need to impose an equivalence relation on them to make this happen. From an algebraic perspective the usual way to solve this problem would then be to use the equivalence relation $\sim$ generated by our necessities (composability, const paths are identities, invertibility, stability under reparametrization…).

Without thinking about this too much I always thought this would be the equivalence relation given by homotopy, until I recently read in this lovely little blogpost that this is not the case. But then

Why is the answer to defining the fundamental groupoid „consider paths up to paths“ and not „consider paths up to canonical requirements“?

Disclaimer: this is a purely pedagogical question, I am not questioning the use of homotopy. But I feel like even if we introduce homotopies before trying to compose paths (ie. if we don’t use paths as motivation for homotopies, but vice versa) it would be a lie to just say „we don’t have to worry about composability, because we already know about homotopies“, if there is another obvious solution…

As always: thank you very much for your time!

TLDR I think it is a quite canonical idea to compose paths. Can this be a hook for developing the notion of homotopy? Does the desire to compose and invert paths naturally lead to considering paths of paths ie. homotopies?