Cool video. The Dominion had the Federation with its back to the wall with their weapons that Federation shields could not stop.
Sisko did not have over all command of the war that seemed to be Admiral Ross' job. Sisko may have been his right hand man on the Dominion/Cardassion Border, Picard could have been his other top commander on another front.
Ross seems to be as close as they came to an Eisenhower type character, the over all commander of the Allied forces. Sisko may have been his right hand like I said due to the stations proximity, but I'm sure many other Starship Captains commanded fleets that answered to Ross. Picard may have been one such Captain.
Picard would have commanded the fleet had Cardassia invaded (Chain of Command). But after the events of that episode he may have been moved to different fleet due to the slight possibility that he did giveaway some hint of information that could be used.
Al in all the war would have not really changed all that much. Picard, may have been able to get other Romulan Senators to reject Senator Vreenax, and convince the Senate that the Dominion was as much a threat to them as to the Federation. Never underestimate Picard's power of persuasion and choice of words. His diplomacy with the Romulans and Sisko making a few key strikes into Dominion space could also have had the Romulan Senate see that ultimately, the Federation/Klingon Alliance was going to win.