A "department of legitimacy" would have to give out a lot of tough love and crack down on delegitimizing behavior, that's for sure.
As case studies I would point out: (1) a sector of Republicans who pursue one kind of legitimacy, "sticking to their guns" and posturing with extreme positions (e.g. they feel like they have at least do something to buy the perception of legitimacy of somebody) and (2) Democrats who take legitimacy for granted (I'm Hillary Clinton, of course I am legitimate). In Hillary's case there was that experience of being on the receiving end of Richard Mellon Scaife's (oddly an unnamed villain in https://www.amazon.com/Zero-Sum-Society-Distribution-Possibi...) war against her husband which convinced her that any illegitimacy perceptions were part of a "vast right wing conspiracy" thus nobody could have convinced her that she could have changed her behavior and won in 2016.
https://www.amazon.com/Zero-Sum-Society-Distribution-Possibi...
talks about numerous problems we experience today, such as windfall profits in the petroleum industry, but the most interesting bit in my mind was the fall of the steel industry in the US which based on the Open Hearth Furnace and the Bessmer process. Japan and Germany had there obsolete factories blown up so they switched to the newly developed Basic Oxygen process which was faster, cheaper and produced superior steel. US Steel lobbied to get protectionism, but this had a negative effect on other parts of US industry. For instance US automakers were making expensive and inferior cars out of expensive and inferior steel.
The problem wasn't that the US Steel industry had to invest in new steelmaking capacity, but rather it had to disinvest in the old technology because so long as it could make profits without investing it wouldn't invest. This character played a role in it
https://en.wikipedia.org/wiki/Richard_Mellon_Scaife
and he later got famous for bankrolling a campaign to harry Bill Clinton when he was in office. Oddly Bill Clinton became friends with RMS later because RMS let him "triangulate", so long as he could keep his head above water he could do nothing for leftists and have them blame the Republicans and not Clinton.
If the phone lines just disappeared it would be perfectly profitable to roll out a fiber network, but so long as the copper wires exist phone companies have every reason to postpone copper retirement indefinitely.
Until then China was more aligned with Russia but Nixon and Kissinger really hit it off Mao and Zhao Enlai and eventually you got Deng Xiaoping who had slogans like "To get rich is glorious" and China went on a path of encouraging "capitalist" economic growth that went through various phases. Early on China brought cheap labor to the table, right now they bring a willingness to invest (e.g. out capitalize us) such that, at their best, they bu make investments like this mushroom factory which is a giant vertical farm where people only handle the mushrooms with forklifts
https://www.finc-sh.com/en/about.aspx#fincvideo
(I find that video endlessly fascinating because of all the details like photos of the founders using the same shelving and jars that my wife did when she ran a mushroom lab)
Contrast that to the "Atlas Shrugged" situation we have here where a movie studio thinks they can't afford to spend $150 M to make a movie that makes $200M at the box office (never mind the home video, merchandise, and streaming rights which multiply that) which is the capitalist version of a janitor deciding they deserve $80 an hour.
This book by a left-leaning economist circa 1980
https://www.amazon.com/Zero-Sum-Society-Distribution-Possibi...
points out how free trade won hearts and minds: how the US steel industry didn't want to disinvest in obsolete equipment which had harmful impacts on consumers and the rest of our industry. All people remember though is that the jobs went away
https://www.youtube.com/watch?v=BHnJp0oyOxs
That focus on winning at increased international competition meant that there was no oxygen in the room for doing anything about interregional inequality in countries.