I'm not familiar with "the" Business Plot, which I guess is a staple trope here.
However I do know of a Business Plot. It happened in real life, in the 1930s.
Fairly early in FDR's first administration, some people approached the recently retired US Marine Corps general, Smedley Butler. Butler was a popular public figure. Most recently, he had helped lead the Veteran's March on Washington during the Hoover Administration.
A group calling itself the "Liberty League" wanted Butler to become the figurehead of a coup. He would be given a new, specially created Cabinet position, President Roosevelt would be sidelined and silenced on "medical" grounds, and in the name of the President Butler would be the actual authority. In turn the coups organizers would be quietly behind him, advising him--everything would go well as long as he took the advice.
Butler was outraged, but he dissembled, hoping to gather enough information to turn over to the legal authorities. He didn't completely hide his first honest reaction--he figured it wouldn't seem plausible that he'd just smile and agree, so he demanded more explanations and assurance, and so he quietly built up a dossier of data on this scheme. A scheme which, by the accounts I have read, included a substantial section of the United State's most prominent citizens, mostly among the very richest.
Eventually he took his scoop to the Congressional committee that preceded the Un-American Activities committee--maybe it already had that name, I believe this was about 1935. And as this was a Democratic controlled committee (the Democrats being comfortably in the majority) and its chair and most members were loyal supporters of FDR and the New Deal, and therefore targets of the Liberty League's scheme, Butler expected the plot to be dealt with handily.
The plot was certainly checked. (And it would not have been easy to find another figure for the strongman face of the new regime than Butler--so his defection (from their point of view--his loyalty to the legitimate government of the USA was his viewpoint!) kind of left them high and dry anyway.
However, all business relating to handling the matter was conducted in closed session, while the committee met in New York City--they had gone there to investigate crime, and while Congressional rules would have forbidden closed session in a regular meeting, for this kind of auxiliary meeting it was allowed.
It was a push. The plotters did not get to go forward with their scheme. But none of them were arrested, or tried, nor even exposed to the court of public opinion. No mainstream newspaper would touch Butler's story--many of them being owned by the plotters.
I think there is your answer right there. If a Business Plot succeeds, people will know about it, but if it fails then mainstream authorities are too entangled in the same social circles and spheres of influence to risk exposing them, since such allegations, even when provably true (or rather, especially then) undermine the whole social order. Thus even people targeted for removal or even death should the plotters succeed often feel that on the whole, it is best to keep it quiet and let it ride.