Honestly though, as much as One Piece is a fantasy manga, Oda does try and maintain some level of real World politics in the story. Sure, the Admirals are licensed to kill, and will execute someone if the need arises, but they aren't executioners. If Kaido gets captured, calling in an Admiral to kill him would not be on the top of the to do list. And even if that was the case, location of where Kaido is captured, in conjunction with the nearest Admiral is key as well.
Kaido doesn't seem to care if people try to kill him, in fact he welcomes it. What he does care about is when people fail to kill him. When that happens, he loses interest, wrecks whoever it was that tried to kill him, and moves on. If he gets captured, and they try to kill him and fail, he's not going to sit around and wait for an Admiral to be called in to finish the job.
Also I believe that the possibility of someone being able to kill him is what's most appealing to Kaido, rather than him actually wanting to die. If he truly wanted to die, he could have just let Oden finish the job. However when the prospect of dying became a reality, Kaido fought to live.
I'm sure this will all be fleshed out in Kaido's backstory, whenever Oda decides to get to it, but there is definitely something more than meets the eye with the whole "Kaido wants to die" plotline.