1) If A is functionally identical to B, then A is a layer of indirection
2) If A is functionally distinct from B, then A is likely an abstraction
3) If A is functionally distinct from B, but B must be considered when
handling A, then A is a leaky abstraction.
The history of mathmatical advancement is full of very grounded and practical motivations, and I don't believe that math can be separated from these motivations. That is because math itself is "just" a language for precise description, and it is made and used exactly to fit our descriptive needs.
Yes, there is the study of math for its own sake, seemingly detached from some practical concern. But even then, the relationships that comprise this study are still those that came about because we needed to describe something practical.
So I suppose my feeling is that, teaching math without a use case is like teaching english by only teaching sentence construction rules. It's not that there's nothing to glean from that, but it is very divorced from its real use.