I think it's important to take a historical perspective. There was a time not so long ago when computers as we know them now did not exist. At that stage, coming up with a precise definition of an algorithm or of a Turing machine was a major advance, allowing one to build the earliest modern computers and begin the revolution that we take for granted today.
As actual computers became more powerful, interest shifted from the computable/uncomputable boundary to the feasible/infeasible boundary, where initially the definition of "feasible" was (roughly speaking) "polynomial time." So then we get the P = NP question and the birth of computational complexity theory as we know it today.
As computers became more powerful and more diverse, interest again shifted. People today are increasingly interested in parallel/distributed algorithms, cloud computing, SIMD architectures, etc. Datasets are so large that polynomial time doesn't cut it any more; people want linear time or even sublinear time algorithms.
So at the time of its invention, computability theory was about practical algorithms. The same goes for computational complexity theory and other subjects in computer science. But as technology advances, the definition of "practical" changes, so that the classical subjects no longer line up so nicely with the interests of current practitioners. That doesn't mean that the classical subjects are no longer of interest, because fundamentally important mathematical concepts never go away. But they become more abstract, and it takes a broad perspective to see their motivation and to be able to tell which problems are still of importance today. For example, in my opinion, some of the most exciting developments in computability theory today are its unexpected connections with differential geometry, as for example described in this paper by Soare. This work is very far removed from "practical algorithms" but illustrates how the study of fundamental mathematical concepts can reap unexpected dividends and is therefore worth pursuing even if immediate applications are not visible.