First, some of the stats of Katakuri you've listed aren't actually superior to King's but that's a debate for another day. The reason why King used speed mode against Zoro is that he realized it can't protect against Zoro's attacks after CoC. That's why he started dodging Zoro after coming out of the basement, where he took Zoro's first CoC-based attack. It was later confirmed here:
So he tried cornering Zoro via speed, which ultimately failed. He doesn't have to do that with Katakuri because Kuri is no threat to him in Flame Mode, unlike Zoro.
Katakuri isn't this battle genius that you're trying to paint him to be. Marco is smarter than Zoro, but he didn't figure out King's ability in what appeared to be an hour-long battle. Zoro and Luffy are generally dumb but their battle IQ is genius-level if not better. Still, I am not gonna deny Kuri can figure his ability out assuming King actually feels it is necessary for him to switch b/w speed and defense modes—Although I don't see why King would even bother using speed mode after witnessing how Katakuri fights.
But still, even if Katakuri figures it out, he doesn't have any attacks to quickly beat King. Let's say he hits him with a strong attack in speed mode, will King still continue to use speed mode knowing he can't hurt him in flame mode? I doubt he is that dumb.