4 ideas
Claudius, grandson of Livia and nephew of Tiberius, stammered, limped and twitched, so his family treated him as an embarrassment and kept him out of public life while Livia poisoned or exiled nearly every plausible heir. After Caligula's assassination in 41 AD, a Praetorian guard found him hiding behind a curtain in the palace, and the soldiers proclaimed him emperor because a harmless-seeming Claudian was the easiest figurehead to install.
In a system where any capable rival is eliminated, appearing incompetent takes a person off the list of threats. Claudius deliberately exaggerated his stammer and clumsiness so that the people deciding whom to kill judged him not worth the trouble. The price is the loss of standing, respect and influence, paid in exchange for staying alive.
People speak and act openly in front of someone they have dismissed as an idiot, because they don't expect that person to understand or remember. Being dismissed therefore gave Claudius access to the court's real workings, and he recorded them as a historian. The person everyone writes off can become the best-informed observer in the room.
When the ruler's first priority is removing threats, the able and popular die young, as Germanicus, Postumus and Marcellus do in the novel. Succession then falls to whoever looked least dangerous, not whoever was best suited to rule. So a system built on eliminating rivals ends up handing power to the survivor who seemed weakest, and it is unprepared for his rule.