The two main reasons I use core lams is first, the osage boards I buy are 5/8" thick and when I rip them into two 1/4" thick lams, I need a core lam or a long power lam to add to the stack to give me the thickness I need at the handle and past the fades. Second, it's easier to get the R/D shape with 3 lams instead of two.
I've used hard maple, walnut, cherry, and red elm for cores and haven't noticed any real difference performance wise between them. They're all in the same ballpark for SG and I would imagine that the tension and compression work is done by the back and belly lam anyway.