Is it believable that Brady and Belichick weren't exactly on the best of terms over the past year or two? I don't think there can be much doubt on that given the amount of smoke we've seen from a variety of sources, including Tom Brady himself (Tom vs. Time comments, skipping OTA's, etc.). Brady has taken less than market value to play for the Pats, and if he perhaps feels a bit entitled, I don't think anyone can blame him for it. He wouldn't be the first superstar to get turned off a bit by the business of football.
Belichick is in some ways in a "difficult" situation as well. The "year too early" comment he made to Cashman was made long time ago and did not reference Brady, but it does reflect Belichick's philosophy; the Pats have either traded vets on their final contract years or let lots of players walk via free agency. Some of those players have continued to be productive, but many higher profile veterans have not. He made a direct reference to Brady's age when he drafted Garoppolo. So it's not unreasonable to believe that Belichick was indeed planning to move on from Brady before the QB turned 41, and that Brady may have felt a bit hurt by that. Belichick's perspective is that he could play the good soldier to Kraft and let Brady play until he's ready to retire, but if Brady were to fall off the cliff in 2020 and the team is 1-15 two years later and son Jonathan is running the team, Belichick would not be immune to being shown the door. I can certainly see why Belichick wanted to keep both QB's on the roster for this season, and I could understand why he may have been upset when he found out there was simply no way Don Yee was going to let that happen no matter what Felger and Mazz and Shank claim to the contrary. It was one case where the salary cap won, and there was no feasible way to simply "work around it".
So, by definition, the interests of Brady and Belichick are not 100% aligned to each other. But here's the reason why all of these articles cause the reaction they do: the authors weave a lot of their own speculation into the story. While they claim to be piecing together nuggets from reliable sources, they seldom make clear what they've heard vs. what they have inferred. Unless you dig in, and then you find out that they've inferred a lot (aka, the "divorce"). And a lot of that inference is often simply unsupported. For example, the claim that the coaches think Brady is nothing special is absurd, and is likely instead a misinterpretation of the coaches simply stating the truth: Brady, like any NFL QB, can be subject to bad games and can be beat. It's easy to see a position coach telling his players "Brady ain't gonna bail us out this week, so I need to see better from everyone in this unit".