There are roads which are twisty enough so you cannot go over the speed limits. Like... a twisty road with 180 degree corners that are driven fast with lets say 30mph, while the limit is 55mph.
On those roads you can drive fast without breaking the speed limit (and without street racing). And you can drive faster with a good handling car. Therefor, in corners, you'd want a good handling car.
Also... a good handling car feels better, and is more fun in corners (and more safe).
Why do you think people started building better and better cars/suspension/tires/etc? Probably not because they wanted to do street racing.
PS: The latest Corvette is supposed to be a great handling car (according to tests). Correct me if i'm mistaken, but i thought that would be an american car...