That sidesteps the question of why all of these comically evil people are okay with using this arbitrary contest to determine succession, instead of the usual route of organically murdering each other until someone emerges who is good enough at disposing of potential assassins that they keep the throne for a while?

I’m saying they’re not ok with it, but they are trapped in the current system. It’s like, an allegory, maaan.

I didn’t watch the movie, so I’m probably missing something. Did Georgiou also have to deal with a bunch of normal assassination plots after officially gaining the throne? Or are we to assume that by virtue of winning this contest, she is widely seen as too dangerous for anyone to attempt to usurp?

Sort of the latter, which is kind of the central plot device.