Ironically properties of groups were one of the first things to come to mind where working on an abstract level makes sense. You can convince yourself groups are an interesting interface/definition (both with a couple concrete examples plus a fuzzy argument that anything you want to call "symmetries" ought to be a group), but then if you want to understand something like commutators and abelianization, I don't really see any insight from working with concrete groups. The point is more abstract: [G,G] is normal (easy to show), so you can mod it out, and doing so gives you "G but everything commutes" (because you quotiented away all the commutators).
Similarly I'm not sure there's a lot of understanding to be gained from writing down cosets (as in the actual list of set members). It's still necessary to know how to calculate, but the understanding of what you're trying to do comes from the fundamental theorem on homomorphisms/first isomorphism theorem. You'll never get it from looking at cosets, and in some way it's actually a bit of a distraction IMO. They're often kind of just there to prove quotients exist.
Tensor products feel similar I think: actually constructing the tensor product is not terribly interesting and mostly just demonstrates that it exists. Focusing on the concrete definition makes it easy to miss the forest for the trees.
Maybe having given a couple examples, we can extend it to the abstract idea that universal properties are usually more interesting than the associated concrete constructions. :-)