He lost enough influence that he lost his role and was not only unable to maintain the military rule, but was unable to maintain any official role within the state despite his attempt to.

Pinochet wasn't the sole ruler, but the leader of a junta. In 1980, the junta announced a referendum would be held in 1988. He lost, and from my understanding it was other members of the junta -- military and police officials -- who frustrated his attempts to ignore the result. I think this was a result of international pressure (the regime kept killing or disappearing foreign citizens, plus the pope condemned the government), the neutering of the socialist movement around the world (the Cold War was ending), and Pinochet getting old (the term he wanted from the '88 referendum would have had him rule well into his 80s).

Pinochet did maintain a role in government, too. He was commander in chief of the army until '98 and a senator "for life" until '02.

Which is a fairly broad definition and you could include many capitalist regimes

This makes it unworkable for me -- what capitalist regimes wouldn't fit? There is no capitalist state where the interests of capital and the interests of the state aren't closely aligned, because capitalism is characterized by the state controlling the working class on behalf of capital. There's occasional state pushback (liberalism), but it's all very gentle and inevitably rolled back.