I agree that there are many natural groups that should be given much more attention when teaching group theory. But the virtue of abstractness to me is in the

I agree that there are many natural groups that should be given much more attention when teaching group theory. But the virtue of abstractness to me is in the notion of isomorphism, which is surely critical to how we understand groups. Then you can find same (that is, isomorphic) groups with different representations - like sitting as subgroups in two different symmetric groups. And to define when two groups are isomorphic is more or less the same as to define what a group is abstractly.