To me, the interest in model categories stems from Quillen's observation that the tools of topology (e.g., CW approximation) can be applied in so many different settings, especially in algebra. But not all of those settings are simplicial model categories. Many of them are dg-model categories, i.e., enriched in the category of chain complexes over a commutative ring. There is a huge literature about dg categories, and they have real applications in representation theory. Using enriched model categories, people have been able to prove that the theory of dg-categories closely parallels the theory of spectral categories, i.e., categories enriched in (your favorite) monoidal category of spectra. This has been a fruitful way to use results in stable homotopy theory to prove things in representation theory, homological algebra, and triangulated categories. It also sets the stage for results about Fukaya categories, homological mirror symmetry, etc.
Results in higher category theory (by which I mean weak $n$-categories, not $(\infty,n)$-categories) also often require enriched categories. At its most basic level, an $n$-category is a category enriched in $(n-1)$-categories. So, many of the models for weak $n$-categories (e.g., invented by Bergner, Rezk, Barwick, Ara, Tamsamani, etc.) are cartesian model categories, so that the next level can be enriched in them. To do homotopy theory in this context, you need enriched model categories. I used them a lot in a recent paper with Batanin proving a strong version of the Baez-Dolan stabilization hypothesis.
Another important reason to care about enriched model categories is monoidal model categories, i.e., self-enriched model categories. Homotopy theory is full of situations where the monoidal structure has been crucial, e.g., in chromatic homotopy theory, K-theory, etc. You probably know there was a long (and eventually successful) search for a good monoidal category of spectra, and then tons of applications. You might be interested in the paper "Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories" by Thomas Nikolaus and Steffen Sagave, that proves every presentable monoidal $\infty$-category is modeled by a combinatorial monoidal model category. Perhaps you'd be interested to generalize this result with "monoidal" replaced by "enriched."
The MathOverflow community has already answered the question "Why do we need model categories?" and "Do we still need model categories?" The same answers tell you that we still need monoidal model categories. And, even if you only want to work in the setting of monoidal $\infty$-categories, you will quickly find that you might actually need monoidal model categories to compute things, in much the same way as described in those answers.
