The OP raises an important point: My own, completely uninformed opinion is that the cheating helped. Ergo, not cheating (too risky for them now were they to get caught cheating in the same way) hurts.
The one constant inconsistency among pundits and commentors who sweep the cheating under the rug is that Belicheck* is a genius, as calculating a thinker to ever walk the sidelines. If he is such a genius, why would he waste so much time and energy on something--cheating--that had no impact on their chances for success?
Either everything he does puts his team in a better position to succeed, including the cheating, or he wasted a lot of time cheating. Which is it?
kj
Adding:
And, respectfully, to MrWEO's point: I read that piece. King stands out among those who are guilty of what I outlined above (and maybe it is only a big deal to me, but I hold grudges.) But look at their record in the playoffs, where they can't fatten up on crappy teams like ours, where teams are more evenly matched in talent and where coaching chumps like Jauron aren't usually found. In those situations, even a small advantage (cheaters) can turn the tide. And again, absent that advantage, the winning percentage would--did--go down.