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The focus on weak equivalences instead of homotopies is largely a consequence of Grothendieck's slogan to work in a nice category with bad (overly general) objects, rather than working in a bad category that has only the good objects. Typically, there is a good notion of homotopies between maps that is well-behaved, but only on the "good objects". If we worked with a category consisting of only the good objects, then we wouldn't need weak equivalences, but we also would be sad because our category probably wouldn't have things like limits and colimits, and would generally be difficult to work with. So instead we enlarge our category to allow objects which are "bad" and which don't directly relate to the homotopy theory we really want to study. To do homotopy theory with the bad objects, we introduce a notion of weak equivalence which lets us say every bad object is actually equivalent to some good object, as far as our homotopy theory is concerned.

A basic example of this is simplicial sets and Kan complexes. Simplicial sets form a really really nice category that is easy to work with combinatorially or algebraically. However, on their own, they are awful for the purposes of homotopy theory. If you model some nice topological spaces as the geometric realizations of some simplicial sets, then most continuous maps between your spaces will not come from maps between the simplicial sets, even up to homotopy. We can define a notion of homotopy between maps of simplicial sets, but it is really poorly behaved (it's not even in equivalence relation, though you could take the equivalence relation it generates).

Now, there is a very special type of simplicial set which is really good for modeling homotopy theory, namely Kan complexes. The singular set of any topological space is a Kan complex. Homotopy classes of maps between two Kan complexes are naturally in bijection with homotopy classes of maps between their geometric realizations. So we have this great theory of Kan complexes which models the classical homotopy theory of spaces and has the advantage that our objects are more combinatorial and we don't have to deal with the pathologies of pointset topology.

However, despite all the nice things about Kan complexes, they don't form a particularly nice category. They aren't just the category of presheaves on a simple little category like simplicial sets are, and don't even have colimits. We can't work with them combinatorially nearly as easily as we can general simplicial sets.

So, we'd really like to use the entire category of simplicial sets and not just Kan complexes. But this is awkward, because we don't have a good notion of homotopy for simplicial sets, and don't even have "enough" maps between most simplicial sets to model what we want them to model. The solution is that we do still have a good notion of weak equivalence which works for all simplicial sets, and after inverting weak equivalences we get the homotopy category we want. Every simplicial set is weak equivalent to a Kan complex, and when working with just Kan complexes, weak equivalences give the same homotopy theory as homotopies between maps would.

Let me end with a more down-to-earth observation. A homotopy between maps $f,g:X\to Y$ is defined as a map $H:X\times I\to Y$ such that $Hi_0=f$ and $Hi_1=g$. Here $i_0:X\to X\times I$ is defined by $i_0(x)=(x,0)$ and $i_1$ is $i_1(x)=(x,1)$.

Now let $p:X\times I\to X$ denote the first projection. Observe that $pi_0=pi_1=1_X$. So, if we formally adjoin an inverse to $p$, $i_0$ and $i_1$ will become equal (both equal to $p^{-1}$), and consequently $Hi_0=f$ and $Hi_1=g$ will become equal.

In other words, imposing the homotopy equivalence relation on maps is essentially the same thing as considering all of the projection maps $p:X\times I\to X$ to be "weak equivalences". In this way, the classical equivalence relation on morphisms approach to homotopy is really just a special case of using weak equivalences. But weak equivalences are more general and flexible, and can be used in settings (like simplicial sets as discussed above) where an equivalence relation on morphisms would not do what you want.