I'll expand on Russell Borogove's answer while essentially disagreeing with the initial thrust of both answers so far, which was about influencing voters who haven't voted yet. Forget all about the potential to influence voters; that's absolutely not the point of calling the election early for Trump Mencken. (Remember, Succession S4 aired in 2023, several years after January 6th, to say nothing of the Brooks Brothers riot of 2000. The writers absolutely understood the 2020 election playbook.)
First, forget the civics class you probably never took in school anyway. Instead, think of the presidential election like any other popularity contest — a Kickstarter campaign, a new club you're trying to start, whatever. The most important thing is to have momentum. If you can get a critical mass early, then it's easier to get others on board. If ATN calls the election for Mencken, it's going to be easier to get PGN to call it, and so on. Remember, the goal is to have everyone calling the election for Mencken. At that point, the votes don't matter.
Everyone's actions here are explicable as attempts to avoid a dilemma, or put someone else into a dilemma.
From Shiv's point of view, suppose ATN calls it for Mencken. Then either Mencken won't have the votes, in which case ATN will have to awkwardly "un-call" the race (an outcome Shiv doesn't want); or he will have the votes (an outcome Shiv certainly doesn't want). So, Shiv doesn't want ATN to call the race for Mencken.
From Mencken and Roman's point of view, if ATN calls the race, that'll put every other news organization in a dilemma: Either they follow ATN's lead in calling the race for Mencken (perfect for Mencken!), or they awkwardly have to disagree with ATN (sowing confusion at best, and looking foolish at worst). At the very least, no other organization is likely to call the race against Mencken; they'll just stay quiet, which means "Mencken wins!" is the only visible narrative.
So, coming out of Election Night, well after all the votes have been cast, we've got the narrative that Mencken has won (and no competing narrative). Why did I say "the votes don't matter"? Well, look at the dilemma again: Either Mencken does have the votes, or somehow he doesn't, in which case we have a huge dissonance between the truth that's been presented to the American people for days/weeks/months ("Mencken won!") and the new story ("Never mind, the other guy won!"). That dissonance, again, leads to confusion at best and riots at worst. You're going to have Menckenites saying that the election was "stolen." The stolen election itself becomes a media narrative. There's no way to put the genie back in the bottle and go back to a world where Jimenez's victory was fair and square and uncontested.
- Which means, from the ballot-counter's point of view, there's probably a lot of pressure to simply go along with the narrative that doesn't lead to riots. ("The ballot-counter" here just means any of the many potential off-camera boat-rockers, from the guy who adds up the numbers to the guy who picks the actual slate of electors.)
The point is simply to make the water outside the boat look dangerous enough that no potential boat-rocker wants to rock the boat.
Meanwhile, regardless of how the vote counts ultimately turn out, what matters is how things look on Election Night. Mencken declares victory: that's it, contest's over. If the votes turn out differently, then we go to the courts and it's a very long and boring process (with all the incentives aligned against the Jimenez faction — remember, one outcome aligns with the media narrative and produces a peaceful transfer of power; the other outcome doesn't).
For more on this playbook, see various speeches by Donald Trump and the other January 6 conspirators in the months before, and weeks after, the election on November 3, 2020. For example, Trump's speech on the evening of November 5:
There are now only a few states yet to be decided in the presidential race. The voting apparatus of those states are run, in all cases, by Democrats.
We were winning in all the key locations by a lot, actually. And then our number started miraculously getting whittled away in secret [...] It’s a corrupt system [...] it’s too easy. They want to find out how many votes they need, and then they seem to be able to find them. They wait and wait, and then they find them.
And you see that on Election Night. We were ahead in votes in North Carolina by a lot — tremendous number of votes. And we’re still ahead by a lot, but not as many, because they’re finding ballots all of a sudden. “Oh, we have some mail-in ballots.”
It’s amazing how those mail-in ballots are so one-sided, too. I know that it’s supposed to be to the advantage of the Democrats, but in all cases, they’re so one-sided. We were up by nearly 700,000 votes in Pennsylvania. I won Pennsylvania by a lot, and that gets whittled down [...]
See, if the initial narrative is that Mencken is winning, then any later attempt to "undo" that win is easy to portray as illegitimate. And once you've eliminated legitimacy — once you're at "It's a corrupt system" — then (1) whoever's got the physical power and mental will to prevail will prevail, in the absence of any legitimate authority to stop him; and (2) a large share of the public will welcome a strongman like Mencken who promises to eliminate the "corruption" and to restore law and order.
TLDR: If ATN calls it for Mencken and the world follows, Mencken wins peacefully. If ATN calls it for Mencken and the world doesn't follow, Mencken wins messily. The only winning move is for nobody to call it for Mencken. ATN and the world are in a big game of Prisoner's Dilemma, and Shiv is desperately trying to get ATN not to defect.