1. Origins

Non-Euclidean geometry was developed without connection to reality.

Axiomatic hyperbolic geometry was a game about arbitrary axioms. Perhaps that's why Gauss didn't publish on it. What he did publish was his work on extrinsic differential geometry inspired by his work as a surveyor. The Gauss-Bonnet theorem for triangles answers a question a surveyor would ask. Riemann said that his intrinsic differential geometry was an attempt to understand space.

Number theory was developed to understand a problem unconnected from reality.

Yes, I think that's what Gauss meant when he called number theory the queen of mathematics. But I think number theory was a lot narrower back then. A lot of things that would now be called number theory were instead grouped under "solving equations." I think that when Abel showed that one couldn't solve the quintic by radicals and when he showed that one could solve it by hypergeometric functions, he thought he was studying the same field.

2. Applications

Group theory and number theory are endemic in CS. It's not just cryptography. Consider coding theory. For an application of 20th C math, Margulis's expanders were for decades the only explicit ones (and one does want non-random expanders for randomness extraction).

The actual importance of finite group theory apart from representation theory appears to be extremely close to zero.

I'm not sure what you mean. Perhaps that when finite group theory turned inwards and tried to classify finite simple groups, it stopped having applications? Maybe I'd buy that. It seems like an argument for the "interconnected" position against the "interesting" position, but fairly neutral for the "applied" position.

Anyhow, this seems to discount decades of 19th C struggle to clarify the meaning of an abstract group, which is clearly important if groups are important. It's hard to see in retrospect what was so difficult. One might credit this advance to set theory, the idea that one should talk about abstract sets (like the set of cosets).

So I think set theory is a quite inward-looking subject that turned out to have great clarifying impact on mathematics. But maybe it's not necessary - do physicists think about groups without it? Similarly, category theory clarified a lot of math, perhaps not in ways that have yet reached the physicists (the way set theory could), but it was been picked up for its own sake in CS, both in type theory and in parallel computation.