Those "battle royales" were the reason for the stability. The process selected for sultans (or, occasionally, mothers of sultans) who were most effective at building a backing coalition, and generally ended in the killing or at least exile of all pretenders to the throne. The disruption the process represented also helped quell the willingness of factions within the government to try and repeat it too often.
The well-established succession processes practiced in the West guaranteed that at any given time, not only was there almost always at least one person who would benefit directly from the ruler's demise, there were often individuals for whom the ruler's premature demise was required for them to be inline for the throne. If you're the King's brother, for example, under male-preference primogeniture you need to make sure your brother doesn't have any kids.
The well-established succession processes practiced in the West guaranteed that at any given time, not only was there almost always at least one person who would benefit directly from the ruler's demise, there were often individuals for whom the ruler's premature demise was required for them to be inline for the throne. If you're the King's brother, for example, under male-preference primogeniture you need to make sure your brother doesn't have any kids.