Here are some rough analogies:
- Model Category :: $(\infty, 1)$-category
- Basis :: Vector space
- Local coordinates :: Manifold
I especially like the last one. When you do, say, differential geometry in a coordinate free way you often end up with more conceptual, cleaner, and more beautiful proofs. But sometimes you just have to role roll up your sleeves, get your hands dirty, and compute in local coordinates. Sometimes different local coordinates are useful for different calculations.
One modern point of view is that Quillen Model categories are the local coordinates of a homotopy theory (i.e. presentable $(\infty,1)$-category, c.f. this MO answer). When you can compute without choosing local coordinate (a model category structure), that is great. But sometimes it can be very helpful (maybe even necessary) to pick a Model structure to do a specific computation. Sometimes it is helpful to jump between different Model structures which describe the same underlying $(\infty,1)$-category.
Model structures are still extremely useful, even essential to the general theory. But they are a tool, just like local coordinates in differential geometry.