In homotopy theory, a typical workhorse theorem is of the form 'there is a model structure on XY with such an such properties'. While there are some standard techniques producing them, most examples need some extra effort. Moreover, even the very definition of a model structure will appear obscure to non-experts and the existence of one does not appear to very interesting in itself.

To be a bit more concrete: One of the most often used examples of a model structure is the Kan-Quillen model structure on simplicial sets, for which there is still no entirely easy proof. It is the basis of any modern treatment of the homotopy theory of simplicial sets. This model structure is also the basis of countless other model structures (like Cauchy-Schwarz produces countless other estimates). Or the various model structures on categories of spectra (S-modules, symmetric spectra, orthogonal spectra), which form the basis of modern stable homotopy theory (unless you want to use $\infty$-categories, where one often uses other work-horse theorems like straightening-unstraightening!).