Jean Luc Picard not wanting to destroy the Enterprise is more than just revenge.He was keeping a promise he made to Kirk , not to let anything getting him off the ship.
The Crew of the Enterprise is the source of divergence into the Mirror Universe.The Enterprise crew was present to ensure that the events of First Contact went off without a hitch. Along the way, they managed to inspire Zefram Cochrane with tales of the future and how humanity manages to get past the post-apocalyptic hell-hole they are currently in. Thus informed, he greets the Vulcans in peace. Now, imagine how Zefram would have been with no Borg attack and no Enterprise crew there: a drunk depressed asshole living in the remains of what was once a superpower nation's nuclear arsenal. When aliens descend from the sky, using a highly advanced starship with who-knows what kinds of weapons to use on the poor, insignificant humans, odds are that he would react in a more wary, more paranoid, perhaps more violent manner (perhaps by shooting said aliens with a sawed-off shotgun and looting their ship) due to fears over what will happen now that highly advanced aliens have landed in the aftermath of World War 3. Thus, the crew Set Right What Once Went Wrong and ensured that the main timeline, rather than the mirror universe, became the 'prime' reality.