I don't get why you keep bringing this up. I have no idea what the Patriots would do.
But just to play your game, I looked at New England's right tackle, Marcus Cannon, who had a fantastic year last year from all accounts. This is what was written about Cannon the previous year: "Backup tackle Marcus Cannon struggled when called upon in 2016. The Patriots' tackle depth was ripped to shreds by injury in 2015, when Nate Solder, Sebastian Vollmer and Cameron Fleming all missed time with injuries. Cannon was one of the first to step in as the team's top backup, but he was beat like a drum, and with almost the same rhythmic consistency."
Cannon had entered the league in 2011. He signed a two-year contract extension in 2014 worth $9 million despite having made just 11 starts in his first four years.
So you have a player who was considered awful in 2015 and was set to make more than Doug Free will this year in 2016. Did Belichick cut him?
Now, I'm not saying this expecting a turnaround from Free like Cannon had. Just that we shouldn't assume the Patriots would cut him if they had him.