It is absolutely not *solely responsible for their success.
I work in the BI Industry - "Big Data" and all that - and once while I was on a flight to Las Vegas for IBM's big BI conference I was seated next to a woman who worked on Belichick's BI initiative. Using the same tools to make football decisions that big businesses use to make business decisions. And she said all of his decisions are based on that sort of analysis, rather than conventional thinking or gut feelings.
And this was BEFORE all of the "win probability" debates started to defend his riskier situations. That seriously increased my respect for him as a manager/decision maker. And when people start to recognize that he always made the right win probability decisions, I understood why, and agreed with him. And that was and is a legitimate, legal, well-earned advantage.
But his pursuit of a legitimate BI advantage comes from the same place as his pursuits of illegitimate advantages. He doesn't say, "How can I increase our chances by thinking/acting outside of the box while staying within the rules". At best he says, "How can I increase our chances by thinking/acting outside the box in ways that, regardless of their legality, will result in punishments that are less than their benefits."
And he cheated against the Steelers. He bolstered his then-fragile authority as a coach, and his then-fragile quarterback situation, by cheating. Removing the question of "reading the defense" - which is especially difficult against the zone-blitz 3-4 the Steelers ran - and replacing it with illicit knowledge of what the defense would do.
There is every reason to believe the Patriots cheated in that game. There is every reason to believe they would have lost if they had not cheated.
There's no way to know how much their success after that season depended upon the fact that they had won the Superbowl AND continued to cheat until they were established as a dynasty. Other incipient dynasties have fallen apart over less than a playoff loss or the loss of a strategic advantage.
After all, the "Patriots' defense is omniscient and Tom Brady is unbeatable in the playoffs" narratives died exactly when Spygate broke.
But I do agree - it would be asinine to argue that Spygate is solely responsible for their success. I will grant that a Spygate approach would not lead most teams to similar success. It was one example of Belichick's relentless pursuit of any advantage, regardless of legality, and the legal advantages have served him just as well as the illegal ones.
But I will say that, if not for Spygate, they would have not beaten the Steelers in 2002. And if they hadn't beaten the Steelers in 2002 - well, who knows? Belichick fired? Maybe not. Belichick forced to put Bledsoe back in? ...maybe.
Belichick sole coaching capital in 2002, and he has invested it very, very well. But he did steal it.