Why is the homotopy category actually important? Since the homotopy category (of whatever) generally has very few limits and colimits, and these don't coincide with homotopy limits and colimits in the lifted model structure, why do we care about the homotopy category? Is it "just" for classifying homotopy types?
 A: As soon as you have a homotopy invariant functor, it factors through the homotopy category, by the universal property of localizations. To give a homotopy invariant functor $\mathsf{C} \to \text{Whatever}$ is exactly the same thing as giving a functor $\operatorname{Ho}(\mathsf{C}) \to \text{Whatever}$. So if we can understand the homotopy category well enough, then we understand homotopy invariants. You know plenty of homotopy invariants: homology, cohomology, homotopy groups... and presumably you know why we care about them. So this explains why we care about the homotopy category.
The fact that the homotopy category behaves badly doesn't mean it's uninteresting. It means that it's hard to understand it. It's because the homotopy category is the "shadow" of something else: the model structure of your category, or however else you want to encode stuff ($\infty$-categories, dg-categories, triangulated categories...).
Think about it like, say, chain complexes and their homology. It's easy to manipulate chain complexes: you can do sums, tensors products, take duals... Without really worrying. But as soon as you take their homology, things start to go awry: you have the $\operatorname{Tor}$-terms that appear for tensor products, extending scalars isn't as easy as just tensoring anymore, taking duals is a mess...
So to really understand the homotopy category, we go one level up and we look at model categories (or any other way of encoding such information). Now things are really nice and well-behaved. Once we've done our work above, we return below and obtain information about the homotopy category, and eventually about homotopy invariants. This is where we get all these spectral sequences and other things, and do concrete computations.
If you look at things now, we already have model categories (or whatever), and so you might think "why care about homotopy categories". But model categories didn't come first: in the beginning, people only had homotopy invariants functors, thought "I will localize at homotopy equivalences" and realized this made a mess of a category. They then realized that it was the shadow of something above it, and they looked for this, and behold, we now have model categories, dg-categories, triangulated categories... But, as far as I understand, this is all with the eventual goal of understanding better the homotopy category, where it's difficult to make computations but where the actual stuff happens.
