A distant Patriots fan
Most certainly there should be a full investigation. Integrity is one of our most valuable assets, as an individual (Bill Belichick or Roger Goodall), a team (The Patriots), or organization (the NFL).
Without a full investigation it brings into question everything the coach has done and is still doing, the teams performance, and the NFL.
If more taping or jamming is found the coach then should be fired. The team needs to have the integrity and courage enough to stand up and take accountability for allowing this to go on. It's like a family's "dirty little secret" where a parent sexually abuses his/her children.
But this time the rape is of the NFL and the family members, the team, are keeping it a secret. The parent in this case is obviously Bill Belichick.
The comparison might sound harsh but it fits. The Patriots need to step forward as a team.