1. When talking or writing about the NFL's handling of the Adrian Peterson case, it's difficult to separate discussion of the vicious cruelty of Peterson's actions from the fairness of the NFL's adjudicatory process. You can wholeheartedly believe Peterson belongs in jail, or in hell, and believe at the same time that the NFL's power is dangerously centralized and wielded capriciously for labor-crushing ends.
2. The NFL recognizes that difficulty, and counts on the inability of the public to divorce the two concepts. It's one thing when the hammer falls on a guy who smoked weed one too many times, but the league knows that no right-thinking person is going to go to bat for Peterson, who stuck leaves in his four-year-old's mouth and whipped his testicles with a stick. It's the boiling frog story, and you tend not to notice the extremity of the heat when it's scalding someone who morally deserves it. But once you've mentally excused the NFL's actions under circumstances as extreme as child abuse, you empower the NFL to use that muscle memory for the next set of circumstances, ones which may not be so extreme.
3. This is a pure PR play on the part of the NFL, and it's almost too cynical to be believed. The league had been reeling from widespread criticism of its eagerness to co-opt the legal process and its inability to sensitively or sensibly handle morality. Peterson—a black-and-white villain—was a blessing. Maybe a bad man, maybe a man who did bad things, he's a relatively uncomplicated figure, and the NFL was thrilled to have someone to position itself against. The NFL clambered over Peterson to regain the moral high ground it never actually deserved, and is using that platform to shout out, "We are strongly against the beating of children." This is the safest and most defensible position in the world. What we're seeing is the return of the soldiers-and-puppies-and-Pinktober NFL, barely months after the Ray Rice fiasco exposed that as a thin facade. There has been no meaningful change. The league is still beyond reproach, because it cares about the children.