The captaincy system in the Bleach universe has three methods of qualification. One can become a captain by 1) getting a personal recommendation, 2) passing the captain proficiency test and 3) winning a trial by combat. Trial by combat is rarely seen in other divisions other than the 11th. But even then it's not exclusive to the 11th Division. Nor is it the only way for the 11th Division to appoint their captains and thus that generation's Kenpachi, it's just the preferred one. So if the captain of the division died of causes that had not been initiated by trial by combat (including those you listed), then the Captain's Council would meet and make use of the other two aforementioned methods. Additionally, an Arrancar killing the Kenpachi of that generation would only mean that said Arrancar is strong. It wouldn't grant him a title or a position in the Gotei 13.

Hope this answers your question and helps you sleep peacefully.

Eldridge out~