Honestly speaking:
You've got a couple things going on with that question. First is the (mostly) false narrative that Belichick moves his best players. He moves players, regardless of skill level, who won't buy in to the program he's running. Cases in point are Jones and Collins just this year. Both were (or had been) great talents, but were either flaking out off the field (Jones) or freelancing and not sticking to the plan (Collins), so they were offloaded. It's pretty hard to argue that the Pats' defense hasn't actually improved since dumping Collins, surprisingly... And he does tend to move certain highly skilled players when he sees a drop off in their skills on the horizon. Logan Mankins is a great example of this. Hell, Randy Moss is an excellent example too - he played his one year, prove-it contract in 07, bought in, and was inarguably their best player. So they re-signed him, he had another good year or two, then just melted down, stopped buying into the program altogether, and was run out of town on a rail. But while he was their best receiver _and bought in_, nobody thought twice about moving him.
But move his best players? Or move all players when their end is in sight? Take a look at guys like Troy Brown or Kevin Faulk. Both dependable players, but there were better folks at their positions while they were around. Tedy Bruschi might fall into that too. All three of those guys retired Patriots, because of the relationships they had formed mostly with the ownership. Look at Gronk - arguably their most dangerous offensive player (behind only Brady, maybe), and even with his injury history, nobody's talking about moving him. Look in the defense at Hightower, Butler, and McCourty. Hell - definitely look at McCourty. One of the best in his position in the league, nevermind the team, and nobody's thinking about moving him.
So we look at Brady. He's another of Kraft's boys - so that's one. He and Belichick are pretty much joined at the hip during the season. That's two. And he's the GOAT (not talking smack here - everyone in New England honestly believes this). They're not likely to move on from him until he's ready to hang up his kleats.
Then look at Garopolo. He has one more year on his contract - so there's no absolute need to make a decision about him right now. But I'm sure there are already talks going on at some level, and they'll heat up once the Pats are done for the season sometime in mid-February. If they think he'll sign at a reasonable level to stay on as a backup until whenever Brady is ready, and it makes sense to the bottom line, he's their best bet at this rate for a Montana-Young or Favre-Rodgers type of transition, and they'll try to make that work. If they don't think they can, they'll move Garopolo now while there's still some trade value to be had and keep looking for their next QB (pretty safe bet that's not gonna be Brisett - I'm still kind of stumped on that signing).