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Isn't that just a restatement of the Peter Principle?

I’d call it more of a close cousin of Michels's Iron Law of Oligarchy
The iron law of oligarchy is a political theory, first developed by the German syndicalist sociologist Robert Michels in his 1911 book, Political Parties.[1] It claims that rule by an elite, or "oligarchy", is inevitable as an "iron law" within any organization as part of the "tactical and technical necessities" of organization.[1] Michels particularly addressed the application of this law to representative democracy, and stated: "It is organization which gives birth to the dominion of the elected over the electors, of the mandataries over the mandators, of the delegates over the delegators. Who says organization, says oligarchy." He went on to state that "Historical evolution mocks all the prophylactic measures that have been adopted for the prevention of oligarchy." Michels stated that the official goal of representative democracy of eliminating elite rule was impossible, that representative democracy is a facade legitimizing the rule of a particular elite, and that elite rule, that he refers to as oligarchy, is inevitable.
. . . which is the basis of O'Sullivan's First Law:
All organizations that are not actually right-wing will over time become left-wing. I cite as supporting evidence the ACLU, the Ford Foundation, and the Episcopal Church. The reason is, of course, that people who staff such bodies tend to be the sort who don't like private profit, business, making money, the current organization of society, and, by extension, the Western world. At which point Michels's Iron Law of Oligarchy takes over -- and the rest follows.