Well, again, I think the pivotal moment in that climactic scene is the one that tends to get overlooked, where Kirk climbed right out on the bridge after he'd almost fallen off it to his death. Most other people couldn't
have done that. They would've hesitated to go back out on that bridge after barely getting off it alive, or would've wasted time trying to find some other way to get the remote, and thus wouldn't have been able to deactivate the cloak in time for Picard to prevent the launch. It took Kirk's special blend of courage, sense of duty, and sheer recklessness to be able to unhesitatingly go back out on that rickety bridge, and that's what made the difference.
Still, I'll concede that the story structure was a bit contrived based on the decision that they needed Kirk to pass the torch. Maybe if they hadn't rushed the film into production, if they'd taken a year off after the end of TNG and refreshed their creative energies before tackling GEN, they might've come up with something stronger. Although I think GEN still works pretty well, all things considered.