Lower doesn't mean more grip. It can mean better handling due to a lower center of gravity, but that's dramatically less important than suspension geometry. That lower cg may not even matter if your roll centers drop so much that the roll moment actually increases.
Stiffness, which most people get only as a side effect of lowering springs, is generally good for grip. This is mostly because of A) controlling load transfer, and B) reduced body roll. For a-arm cars A is much more important, since they gain camber throughout their suspension travel. For strut cars like ours, But matters about equally as much as A.
Unfortunately, in our cars, the suspension geometry, which is the number one priority for making grip, doesn't work well with more than moderate lowering. Once you go beyond that, the lowered cg does nothing to help you. Ideally, there would be a way to adjust our roll centers, but nobody has come up with a good solution for that short of custom LCAs.
My point is that when you say a stock height car shouldn't be able to take turns quicker than you, you're basing that on the idea that low is fast, which is a myth. It's actually quite expected that a stock height car would turn faster than a significantly lower one.