Of course, having there be so many survivors that all the main cast makes it to safety would seriously undermine the "Jem'Hadar message" of ruthlessness...
I personally enjoyed the use of a Picard-like captain in a ship like Picard's more than I would have enjoyed a TNG crossover here, for the cleverness of the dramatic choice. But I'm sure the climax could have been pulled off as a crossover, too. It would just have been different in more ways than the obvious ones.
...In terms of alternate dramatic takes, how about it really being Picard here, but Sisko's XO having been Ro from the get-go? The mini-conflict that endears us to / alienates us from Keogh would have been played out, only with the audience now knowing the full story, and both sides of it.
In any case, at the time of the episode, the audience was supposed to be thinking of the Galaxy
class as the most potent Starfleet combatant; there were no alternatives being offered. So the choice of ship type in the episode would have been dictated by these concerns alone, even if one forgets about this being the hero ship of a sister show. (Although of course the Galaxy
being a top combatant is a direct consequence of it being the hero ship of the sister show.)