A higher spring rate will result in less body roll, it won't necessarily result in improved cornering. It might feel firmer and more precise but it won't necessarily offer more grip or a better balance of grip front/rear. You could go through many different spring rates trying to achieve front/rear balance. Even then, without adjustable dampers, the new spring rates will be unlikely to be matched to your damper rates.
On dry pavement you would have to corner very aggressively to discover these mismatches (like during an avoidance maneuver) but on rain-slickened pavement it could show up easily during a bit of spirited driving. Here in the rainy PNW I see the "ricer" cars spun off the side of the road on a regular basis. These mismatched suspensions were more common back during the years of easy money (2000-2007). Now a lot of those ricer cars are in the junkyard and I don't see it as often.
Typically these kind of suspension mods make the car feel like it has more control but result in slower lap times and worse avoidance maneuver capability. Sometimes the owners then compensate by replacing the tires with a sportier, grippier compound which helps mask the issue until colder weather arrives. But most people just want eye-candy and the illusion of sporty control and don't know (or don't care) if the performance (and the safety) of their car has actually decreased. Suspension tuning is a real art/science and true experts are few and far between. When damper rates, spring rates, etc. come together the car will have exceptional manners and be easy to drive near the limit, especially on slick surfaces like snow, ice and cold rain (and Mazda has done an exceptional job of tuning the CX-5). Sure, dry road handling and grip can be improved by throwing ride comfort and rough road performance out the window but it's not as easy as simply slapping on some shorter springs with higher spring rates because then your spring rates are no longer matched to your damper rates.
So, when you say "improved cornering" you are making a lot of assumptions.