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 Dolbeault cohomology and coherent sheaf cohomology in particular, I would say the premier applications are to classification-type problems in algebraic geometry. 

First, Dolbeault 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 Dolbeault 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 étale cohomology.