I think the majority of politicians in Western countries have at least one filmed child sex episode. Assume you're a very powerful politician and you need underlings in powerful positions, which you can help novices get, but once they get powerful they become independent and can stop being underlings. Filming them having sex with an kid, or at least an obviously underage teen, is the single best blackmail material possible. Damage potential basically equal to murder evidence, but there's no hard to hide murder victim - and in child sex one victim can be used several times, as opposed to just one time in murder.

As every such cabal would be much more effective than a lone politician, eventually power would entirely shift to competing cabals bound by child sex blackmail - making it impossible to dismantle externally, because both the executive, the judicial system and the biggest media would be controlled by people in such arrangements. Media in particular would have the main job - in case of any genuine leaks, mix the truth with outlandish claims to discredit it as a dumb conspiracy theory.

The game theory works out so nice I don't see any alternative, it's the stable state of the political system. I can see it being murder though, if disposing a body is much easier than my estimate.

In non-Western societies it could be something else. In North Korea, being filmed shitting on a photo of the current leader would work much better I guess.