At this point, Janeway knows next to nothing about the Borg. Not unlike Picard, who had much more on-hand experience with them. Sure, by the end of the series it was the other way around, but at the time of Scorpion... no. Picard would know better then to trust the Borg, and would never even consider an alliance. He would find other ways to get to the core of the problem, perhaps even open up a dialogue with 8472.
I wonder if Janeway, knowing what she learned of the Borg by the end of the series, would be so open to the idea of an alliance with the Borg.
How do you open up a dialogue with a race whose goal is to destroy everything in your galaxy. The only way that one was established was because Voyager and the Borg discovered how to defeat the bioships of Species 8472.
Picard is not stupid, and he may have been hurt nearly beyond repair, but he knew how to compartmentalize his feelings.
1. Species 8472 is out to destroy all life in the galaxy.
2. Find a way to stop them.
3. Our idea will only work with Borg help.
4. Ally with the Borg.
5. Deal with Species 8472.
6. Cut and run from the Borg.