This is just my personal headcanon, but I like to think that the biggest reason Kaigaku couldn't get along with Zenitsu is that he saw alot of his own weaknesses reflected in the younger male. Zenitsu reminded him just how weak willed he really was even after all that training and he didn't like that. So, he lashed out.
I also like to believe that he really did think of him like a little brother of sorts. But damned if he'd ever admit to it...
