Con No. 1 - The grass is not always greener
So, let's say you are one of the Patriots fans who just think there needs to be a change at both GM and HC in New England. That is fine as long as whoever the Patriots replace Belichick with is actually better than Coach Bill. Because otherwise, what is the point? Just replacing a GM and head coach doesn't truly fix things.
How many teams in different sports have made changes, and then those changes turned out to be for the worse? Much worse. Just ask current Patriots linebackers coach Jerod Mayo. He played at the University of Tennessee under long-time coach Phil Fulmer but Vols fans wanted Fulmer gone and eventually, he was. Tennessee then struggled for more than a decade. The same could happen to the Patriots if Belichick is let go.
Pros No. 2 - Sometimes a fresh set of eyes are needed
Let's forget about the above for a minute. We can assume that replacing Bill Belichick actually does work out. For whatever reason, maybe just a lack of seeing the bigger picture of overall changes needing to be made and seeing things from only one point of view for decades, an NFL team can become shunted from adjusting to an ever-changing league.
Heck, when Bill Belichick became the Patriots' head coach, Seattle Seahawks 14-year head coach Pete Carroll (Patriots fans remember him, right?) was in his year off from coaching (after being let go in New England) before he became the head coach at USC.
Now Carroll is the oldest coach in the league, and Belichick is the second-oldest. This is not an agist bit, but Belichick is not your guy if a team wants to have a coach, general manager, or a combination of the two for the next ten years. Neither is Carroll. Maybe a Sean McVay-type would do the trick in New England.