If Dax had become male, that scene where Ezri and Worf were stranded on a planet after she went looking for him and they wind up in each other's arms might have been played quite differently.
They probably could have used a third female character from the beginning, but the problem would have been where to put her. Odo? Then both main Bajoran contacts would have been female. A female Quark wouldn't work if females weren't allowed to participate in business. A female Starfleet doctor would have been a repeat of Beverly. O'Brien was there to pull TNG fans. That leaves Sisko, and TPTB weren't ready for a female leader yet.
I still think the bettef answer would have been to elevate the status of Yates. She did work for the Bajoran government. She could have been given responsibilities that reflected the planet's role in the war (or its response thereto).
One problem I have with saying that Ezri was a horrible character is that DeBoer was a much better actres than Farrell--they were at least equals at doing light comedy/romance.