The common use of that termis to describe administrative agencies with regulatory power, independent of whether they are independent agencies; its a (hostile and derogatory reference to) bureaucracy distant from elected officials, not a theory of the positioning of independent agencies in contrast to other executive agencies.
Citation needed. My reference goes all the way back to 1937 and is specific to independent agencies.
> Almost fifty years of experience has accustomed lawyers and judges to accepting the independent regulatory commissions, in the metaphor, as a headless 'fourth branch' of government.
The full paragraph you're excerpting from Wikipedia is much less convincing:
> Such groups can include the press (akin to the European 'Fourth Estate'), the people (in sum or as grand juries), and interest groups. The independent administrative agencies of the United States government, while technically part of any one of the three branches, may also be referred to as a ‘fourth branch’.
Anyone using the expression to refer to "the people" or "the press" obviously isn't talking about those groups' legal relationships to the de jure branches of government. My sense (from the Wikipedia article, but also anecdotally) is that "fourth branch" is just a colloquial way to refer to various sociopolitical groups. So I don't see this as a convincing rebuttal to dragonwriter's comment.