Let me try to turn this question around in the following way: we need model categories since we do not really understand why such a perfectly symmetric structure can capture the very essence of homotopy-related phenomena.

When I first saw axioms of a model category I was at the same time awestruck by breathtaking beauty of the Quillen's masterpiece, and puzzled by the obvious fact that this seemingly purely aesthetic conceptual structure is very efficient at describing lots of specific situations where we have the notion of homotopy in various very general senses.

Let me mention one particular aspect of model categories that has been, and still remains, especially baffling for me. In most typical examples, cofibrations are, at least "morally", monomorphisms, while fibrations tend to be epimorphisms. But in the closely resembling structure of factorization system it is exactly the opposite - the left halves of the factorization are presumed to "behave like" epis and the right ones like monos.

There are several other mysteries related to model categories. Let me mention just one more. Along with Grothendieck's derivators, there are several closely related structures developed by Franke, Heller and several others. They suggest that a model category is actually the tip of an iceberg, encoding a flood of structures derived from it. I agree with the opinion that from modern viewpoint thus it is more natural to switch to $\infty$-categories; but is it yet well understood how a model category structure encodes all these higher order structures so efficiently?

So my answer is - we need model categories to understand what they are trying to tell us about homotopy theory.