"Descent", that is all the proof I need.
As for the crew and Picard, it was just bad writing they never considered the possibility of it failing.
Was Picard wrong? No, you don't commit casual genocide. It would be NO different than if Kirk unleashed a virus to kill all Klingons/Romulans in TOS, or if the Feds allowed the virus to kill all Founders in DS9. You want Picard to do that to the Borg, then you agree with other Captains doing the same to their enemies as well.
And no, the Borg being a Collective makes no difference.