Unlike what fantasy booking AEW trannies believe, you can't just "fire" and replace someone on the spot. That's not how corporate America works. If Dwayne had the power to fire Paul, he would've already done so and replaced him with Gewirtz. However it's not really that simple. First off, Dwayne's position on the board is entirely irrelevant. He's little more than a figurehead without any major shares. "The Final Boss" is a wrestling gimmick. Second, if Ari were to fire Triple H, the second in command of one of TKO's subsidiaries, TKO would have to pay him a hefty severance. And why the fuck would they do that? Profits are up massively. Unlike what those trannies believe, an incredibly successful business man and shoot tribe member like Ari doesn't give a single fuck about the friendship between Dwayne and Nick, nor do we even know how the relationship between Paul and Nick is. They seem pretty tight to me. Of course since Ari (and Mark Shapiro) own the majority of TKO's shares and are its CEO and COO respectively, they could theoretically demote Paul to a lesser position, which would go over the board where Dwayne WOULD potentially have a vote. However there is no world where any of this is in the interest of TKO as long as profits continue being up.
The reason for the politicking (from both sides) is that both of them want to hemorrhage public support to put pressure on shareholders, but even if the public turned against Paul the chance that he'd randomly get removed is near zero. It's simple business, and all these people are making money for each other, and at the end of the day that's what counts.