Let's not forget to go back to the source for what Leyton's aims actually were. That is, let's assume he was being at least halfway honest with Sisko and trying to convince him to come to the side of the plot whenever the hero officer questioned him or held him at gunpoint.
After Sisko's arrest on false grounds:
Leyton: "The President won't be making a speech. I will."
Sisko: "I assume you'll be explaining to the public why it's necessary for Starfleet to seize control of Earth."
Leyton: "Temporarily, of course. Until the changeling threat has been neutralized."
Sisko: "And how long will that be? Months? Years? Decades?"
Leyton: "However long it takes. The people deserve strong leadership. Someone who can protect them."
Sisko: "In other words, you."
Leyton's aim to take factual charge of Earth's security is explicit there, even if he doesn't say "yes" to Sisko's last challenge. But he doesn't say what sort of leadership he is aiming at. Is he going to depose the President and explicitly alter the form of government? Or just convince the President to take orders from Starfleet?
After Sisko breaks out and confronts Leyton in his office:
Sisko: "Admiral, do you realise what's going on here? Even if you win, even if you do manage to oust Jaresh-Inyo, you still lose. We all lose."
Leyton: "I can't say I agree with you."
So perhaps he isn't going to "oust" the President after all?
Sisko: "Do you think other Federation worlds are going to sit back and let their President be replaced by a military dictatorship?"
Leyton: "Hardly a dictatorship, Ben."
So perhaps Sisko is jumping to conclusions?
Sisko: "Overthrowing a legitimately elected President and giving Starfleet direct control over the government? It sounds like a dictatorship to me, and I'm sure I'm not the only one who thinks so."
Leyton: "There will be some dissenters at first, but they'll fall in line once they realize that strengthening Earth is the first step toward strengthening the Federation.
Well, if Leyton isn't going to throw the President in a dungeon, this would be a good time to tell Sisko he's dead wrong about what's going on. But he never really confesses to what he is planning. Might be the propaganda campaign is all he's ever gonna do - culminating in peacefully convincing the President to let Leyton give a rousing speech and then start acting as Jaresh-Inyo's top advisor and de facto master.
Then, after the space battle, comes the scene where Leyton considers making a fight of it despite being exposed. But that may be his Plan B now that the more peaceful Plan A failed. Indeed, he appears to describe fighting as an alternative
to what he has been doing so far.
Leyton: "It's not over! I have enough loyal officers to make a fight of it."
Basically, then, we have dialogue where Sisko believes Leyton will replace the President as the top leader of Earth, the Federation, Starfleet, or all of them. And where writers probably write with the intent that Sisko is right. But there's plenty of wiggle room for a Plan A that makes good longterm sense for Leyton and stands a chance of success.
That is, everything done so far might have been merely in order to get the President to lend his ear (does he have those?) to Leyton, and Leyton alone. Nothing would change except the policy that Jaresh-Inyo will pursue from that point on. Remember, up till this point, Leyton has done things specifically to persuade and convince the President, in addition to doing things to persuade and convince the general public. This would be wasted effort if the idea were to discredit Jaresh-Inyo and throw him down from the throne while the crowds cheered. Letting the President reign would be a sustainable approach, and also consistent with all the earlier actions.