Just an FYI...
In the last CBA, they changed some rules on trading players that are franchised. It's not against the rules, but the player, players union and Goodell have to all put their stamp of approval on the deal. So it makes it much harder to get a deal done. The players union complained that it takes away free agency from the player and goes against the spirit of the rule. They were going to make it against the rules, but instead they just require consent from everyone.
What will happen more often is a player gets franchised and if something magical doesn't come up and the team can't find a way to sign the player or make a trade, they rescind the tag. That's what happened with Josh Norman last year.
Belichick might want to keep Jimmy because Brady has communicated that he won't play for long. But if he believes Brady will be around for 3+ years and he wants to get something out of Jimmy G, he's going have to move him this offseason.