What a fun question!
I'd like to first mention a speculation on my part: When most people think of $\infty$-categories, the categories they think of probably exist in another formulation. For instance, the category of spaces (or Kan complexes), of chain complexes, of commutative DGAs, of spectra, et cetera, all have well-developed theory. And we're used to doing computations in these categories based on approaches that predate quasi-categories or Lurie's HTT or Higher Algebra. For such well-studied categories, I think you'll almost always find some pre-quasi-category (e.g., model categorical) "computation" that'll get the job done. There's no need to think of them as weak Kan complexes first.
(To compute a homotopy fiber of a map, for instance, we'll probably just replace the map by a standard fibration and compute its fiber thereafter. This is a silly example, but one in which I think working with a model-categorical framework is easier. This is much faster than proving that some over-category has a terminal object.)
But it would be a lie to say that most "computations" one needs to perform in an $\infty$-category can already be done in a non-quasi-category world. Here are two examples:
(1) If you're working with a new $\infty$-category for which someone hasn't done the model-categorical prep work for you. Depending completely on your math-path, you might come across (or define) a category which is most naturally defined as a weak Kan complex. One degree of separation away, you might define a category which is most naturally enriched over Kan complexes, and apply the simplicial nerve construction to obtain a quasi-category. But if you're just a tramp like me, suddenly face-to-face with a new category with some homotopical flavor, you may not have a natural candidate for a model structure, nor have any intuition for how to prove that something really is a "homotopy fiber" for some map, or more generally a homotopy (co)limit for a homotopy coherent diagram. (Whatever a "homotopy (co)limit" means for your category.)
With such a new category, all you a priori have is whatever led you to define this combinatorial gadget (a simplicially enriched category, or a quasi-category) and maybe some interpretations of your morphisms depending on what motivated your definition. So when trying to prove that some object is the homotopy fiber for some morphism in your quasi-category, it might be easiest for you to simply prove that an over-category has a terminal vertex.
I should admit that, while I imagine that examples like this will come up more and more, the only example I have in mind comes from joint work with David Nadler, where we compute kernels in a category we define, which happens to be most naturally a quasi-category.
(By the way, if someone has techniques that make it very easy to prove something is a limit of a diagram in a fibrant simplicial category, please post it as a comment to this post! I'd love to know more techniques.)
(2) If the diagrams you're working with are homotopy coherent but hard to make sense of at a level of strict commutativity. Another example I've come across is to prove that two homotopy coherent diagrams have the same homotopy colimit. Again because there is a concrete model for over/under categories in the $\infty$-categorical model, I could write down an equivalence between the over-categories associated to the two diagrams.
If the non-$\infty$-categorical computations are made possible by model-categorical ideas (like knowing how to replace morphisms), then $\infty$-categorical computations are possible because you reduce your computations to simplicial-set ideas, often proving that a simplicial set is contractible or that it has a terminal vertex. In some mathematical universe both paths may amount to the same thing, but it's my impression that the former approach is hard to follow if you're working in a new category without model structure, and is also not as easy to work with when the higher homotopy coherences of your diagrams are subtle. (Please feel free to let me know if I'm mistaken on this point, I'd love to hear more views.) The quasi-category framework allows you to avoid much of that difficulty by passing the buck to the geometry of simplicial sets.
By the way, you also mentioned something about computing spaces of $k$-morphisms. I have no idea how to do such a thing in the $\infty$-category world, or in either world, really. (Please educate me if someone does know how.)