Yes, you did. You just haven't fully realized it yet. Kaidou was gunning for Whitebeard but Shanks stopped him. When attacking Whitebeard was no longer feasible Kaidou pulled back. He wasn't there for Shanks and had nothing to gain from instigating a full-throttle confrontation with his fellow Yonkou.
Kaidou's life doesn't strictly revolve around suicide. He wants to
unleash a massive war and is
building an army to do it. Kaidou also enjoys some of his islands, like the one
X-Drake attacked. All these are benefits he has earned thanks to running his own personal empire, an empire that is built upon the formidable military power he owns.
Again with this nothing to gain. What was he going to gain from battling WB, who was not only another Yonko but was tagged the strongest man in the world? You're implying kaido was facing some kind of loss if he'd fought as compared to fighting WB. Kaido wasn't after WB's status or whatever. He wanted a fight, a good one and maybe to die. Instead of WB, he got shanks. Might not be what he came for but shanks is obviously a worthy opponent. Even if he doesn't consider shanks as one, it doesn't matter. He's supposedly violent by nature and he's someone that has been captured more than a dozen times by Yonko/marines. That's goes to show how reckless he really is. You're acting as though he's this calm level headed individual who thinks things through. It's contrary to the manga's allusions thus far.
His life might not revolve around suicide but that and violence are clearly a big part of it. His empire is made up of a bunch of bloodthirsty and heartless bunch, even by pirate standards (at least going by the little we've seen). They're equally violent as well.
What's your point with the war? It's just another avenue for him to engage in violence which is consistent with his character and what I've said so far. Building an army doesn't mean much here. Kaido is suicidal but loves a good fight apparently. He can't possibly partake in a war on his own and even his with his crew, it wouldn't suffice. WB came in full force with his allies and was also aided by Luffy and the Impel down inmates but it still wasn't enough. Besides, Kaido has been captured numerous times. Getting captured means being in prison, which means no action. He obviously wouldn't be keen on going to a war with adequate forces only to get captured.
Kaido having an island is irrelevant. I'm not seeing the point of that. He's a Yonko, he has tons of subordinates under him and has territories including numerous islands. Why is it weird that he'd had a favorite and what does that have to do with the matter at hand?
