The owners didn't exist. The Hesses were selling the team, and Woody Johnson bought it two weeks after Belichick quit.
My understanding is that after the Cleveland debacle, Belichick had zero interest in entering an uncertain ownership situation again. He knew Kraft from his year in NE. NYJ ownership was unsettled. I don't think he was crazy about being in Parcells' shadow, but what was happening at the top of the organization was at least as important from what I can tell.
https://www.si.com/nfl/2014/10/15/bill-parcells-excerpt-bill-belichick-new-york-jets
And Woody has proved to be what, if not an amiable dunce and faux tough guy?
To be sure, there was no proof of it at the time, but BB wasn't taking the chance. Just as Parcells did not take the chance -- he elevated himself and found a fall guy in Al Groh, if memory serves.
Parcells himself admitted to this later in his memoirs, albeit in a different context. He acknowledged being a restless soul forever ISO greener grass. He pleaded guilty to "Hamlet like endings." And you know what he attributed his crash landing in Miami to? A "neophyte owner". In other words, what he tried to foist on BB in NY.
The NFL has never known a better turnaround artist than Bill Parcells; he proved it too many times in too many places. And as a game coach, he consistently got the better of Joe Jackson Gibbs, who was about as good as you can get.
But with Parcells, fealty and accolades were supposed to run mainly in one direction. His. That was the case with owners and subordinates too.
So to answer Instaface's question, yes this son of Englewood proved it in many places, but mainly with the NJ Jets. With that head coaching ruse with BB that Tagliabue did not buy for a second. And with the effort to test run Woody as owner with a BB/Al Groh stand-in.
BB, of course, had a front row seat to this stuff.