[[WMG: Jean Luc Picard not wanting to destroy the Enterprise is more than just revenge.]]
He was keeping a promise he made to [[Film/StarTrekGenerations Kirk]] , not to let anything getting him off the ship.
[[WMG: 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 SetRightWhatOnceWentWrong and ensured that the main timeline, rather than the mirror universe, became the 'prime' reality.
* This is presented exactly that way in the beginning of [[StarTrekEnterprise Enterprise's]] "In a Mirror, Darkly" two-part episode: the Vulcan ship lands, the Vulcan greets Cochrane, and then a refilmed version of the scene shows Cochrane (or his hand double) drawing a gun and shooting the Vulcan before the rest of the crowd rushes forward to loot the ship.