You are only in the drivers seat if he other side feels you are. There is zero evidence that the league feels that way. Absent getting the "ball boys" to willing sign a statement under oath (possibly backed up by a lie detector test [for PR]), I'm not entirely sure which side has an advantage.
Starting fresh will require a court order, multiple months from now (at which point we start completely over) - this may only apply to a Player (Brady). There's a value to the resolution of a conflict, however, even with complete vindication we're just back to January (minus Revis, who I believe is tangentially related).
I don't see an end game from Kraft's point of view. Brady might be able to go to Doty and get the Dope's power to Investigate/Punish/Hear Appeal within the CBA greatly reduced ...... I don't know if that's possible or if that would be one of their goals. What would each of Brady and Kraft consider as an end game win.
It all comes down to the "ball boys", can they withstand an investigation under oath. If they can, IMO, go all in and go straight for the throat. If not, IMO, we settle.