This was the first encounter the Federation had with the Borg, so Picard had no idea this was a foe bent on assimilating all of humanity (and everyone else too). They are not a military organization and their mission is exploration, not to destroy species they come into contact with. Quoting from the Season 4 episode, 'Galaxy's Child', after Picard gives the order that kills the space creature they have just encountered (emphasis mine):
We're out here to explore, to make contact with other lifeforms, to establish peaceful relations but not to interfere... and absolutely not to destroy. And yet, look at what we have just done.
Also, as you say, the initial round of phaser fire inflicted significant damage to the Borg cube. The Enterprise crew and Picard would have no way of knowing the Borg's capacity to generate, their ability to adapt to the Enterprise's weapons, or their lack of interest in any sort of peaceful resolution.
It's also a small possibility (pure speculation) that since this is another encounter with Q, Picard does not want to simply start destroying any being they come into contact with. It was not so long ago that Q put Picard on trial for the crimes of humanity. I doubt Picard wants to give Q additional reasons to claim humans are barbarians bent on creating war wherever they go in the galaxy.
to seek out new life and new civilizations? Not destroy everything they see?