But a lot of the reason it's useful/interesting that say sorting is n log n is because we have computers to actually implement sorting. I'm not sure the problem was even posed before computers.
Mathematicians have a long history of being concerned about computational efficiency. Two examples that come to mind are FFT and Euclidean algorithm. I'm certain there are many others.
Before electronic computing machines pretty much took over, "computer" was a job title/description. The exciting part of Turing's On computable numbers, with an application to the Entscheidungsproblem was his abstraction of what computers actually do to a simple machine. Large chunks of what's currently being explored is in relation to problems that we may or may not be able to solve using machines we're not even sure can, in principle, be built, and on that level "computer science" assumes spherical cows in a frictionless vacuum. It's nice that we have relatively compact, fast and nearly ubiquitous machines with which to apply some of what's been discovered along the way, but fundamentally "computer science" is the mathematics of process.
But note that the term "algorithm" is much older than automatic computers; speaking of the problem of sorting, it also had existed for many years prior in the form of various playing card puzzles, but not only - one famous example being Tower of Hanoi.