Why study Higher Sheaf Cohomology?

I think these kinds of lifting-based statements are the wrong place to start. For me these arguments about lifting and such are most convincing as answers to questions like "What are cohomology groups?". For instance $H^1(X,\mathcal O_X)$ is the tangent space to the moduli space of line bundles on $X$. But that don't really explain why to study it. How much do you really care about the Mittag-Leffler problem? Probably not enough to base your entire mathematical career around it. But a huge fraction of modern mathematicians have based their mathematical careers around studying cohomology or generalisations of it.

Instead, as you suggest, what motivates the study of cohomology are its immensely powerful applications. For Dolbeaut cohomology and coherent sheaf cohomology in particular, I would say the premier applications are to classification-type problems in algebraic geometry.

First, Dolbeaut cohomology groups provide natural invariants (the Hodge numbers) that can be used to distinguish algebraic varieties.

Second, the machinery of sheaf cohomology is crucial in answering all sorts of geometric questions about line bundles and other geometric structures. A good example might be the proof of the Hodge index theorem via Hirzebruch-Riemann-Roch, as in Hartshorne. We need all the cohomology groups to define the Euler characteristic, and the Euler characteristic is such a nice invariant that there is a simple formula for it, and this simple formula helps us understand purely geometric questions about intersection of curves.

I would say that at a typical algebraic geometry seminar the majority of the talks involve higher sheaf cohomology at some point, but most talks are not devoted to answering a question expressed in terms of cohomology, so one could find many examples there.

Third, there are the applications via Hodge theory. The natural isomorphism between the singular cohomology and the Dolbeaut cohomology itself has nontrivial structure which is related in a highly nontrivial way to the geometry and arithmetic of the variety. In particular it is supposed to tell you about algebraic cycles - this is the Hodge conjecture. While that is open, many interesting facts are known - e.g. the K3 surfaces are completely determined by their Hodge structure.

Studying how these Hodge structures vary in a family of varieties leads to all sorts of interesting theory, which again can be used to solve purely geometric questions - the statement I remember off the top of my head is that varieties of general type cannot have a nowhere vanishing one-form (Popa and Schnell).

And this is not even getting into the very interesting applications of other cohomology theories for algebraic varieties, such as etale cohomology.

I think you're absolutely right that the function $(i\in \mathbb N)\mapsto $interestingness($H^i$) is a rapidly decreasing function. I heard that Gel$'$fand compared it to the successive derivatives of position: we care about speed and acceleration, possibly about jerk, not so much about jounce or beyond.

However, you very often have to have those $H^i$ if you want to define Euler characteristic (in whatever context), so even if you don't want to compute a particular $H^i$ you want it around theoretically.

In the context of coherent sheaves, very often the approach is to compute Euler characteristic, give some reason that higher sheaf cohomology vanishes, and conclude that one has computed $H^0$. Serre gave a talk (around 1998) in which he explained that "It was once believed that the only good sheaf cohomology is dead sheaf cohomology" but "nowadays this is not the [politically] correct view."

I can't answer the question strictly within the framework of algebraic geometry, but in representation theory of semisimple algebraic groups there is a partial answer: sheaf cohomology groups of line bundles on the flag variety $G/B$ afford natural representations of the given group (as in the theorems of Borel-Weil and Bott), and in prime characteristic the higher degree cohomology sometimes leads to mysterious new representations larger than Weyl modules but still having a unique highest weight with multiplicity 1. It's still an open-ended problem, but the small rank case of $G_2$ illustrates it well: see for example a recent paper by Andersen-Kaneda.

Since 1977 Henning Andersen has found many interesting patterns in higher cohomology, which so far agree with my suspicion that these results are determined by Kazhdan-Lusztig theory (involving inverse Kazhdan-Lusztig polynomials) for affine Weyl groups. See for example an old paper on cohomology of $G/B$ in characteristic $p$ here.