Well, on my home planet of Neecknaw, this is a simple affair. The new leader chooses himself.
Or herself. But we haven’t had a female leader since the Gloomy Ages (the interstitial period between the Dark Ages and the Time of Light) when the orangutan giantess Slothia sat on the preceding and diminutive warlord, Marmostak the Mighty. (Marmostak the Mighty Small the followers of Slothia called him.)
But you’re probably asking because of the upcoming so-called “elections” you are holding in the “democracies” of the large landmass you call “North America”. I think the political theatre you have invented is quite fine, actually, though it lacks a certain martial elan that we on the Planet Neecknaw like to see in our leadership hopefuls.
So in that spirit, I would like to suggest that in lieu of your “elections”, you should have some kind of television show in which the leaders of your political “parties” eliminate one another in gruesome (and entertaining) ways. Because you humans value guile and low cunning as much as the simian population of Neecknaw, I’d recommend something that compensates for brute strength alone — I’m open to suggestions in the comments. (Otherwise, I’m pretty sure Elizabeth May will have an unfair advantage over the other sissified leaders of Canadian political parties, and this should also make the next US election more interesting to watch too, though they are already pretty bloody.)