This claim (from the article) is nonsense. I've changed the behavior of my kids countless times with punishment.
I realize that the attitude I am supposed to take is that when someone with Ivy League credentials espouses a currently fashionable theory that my own observations contradict, I am supposed to believe the theory and not my own lying eyes or I'm "anti-intellectual". Credentials over evidence.
I'll take the risk. But I'm still interested in these theories, because I'm always looking for better ideas to try. A theory that is false overall can have useful parts. Maybe "punishment won't change behavior; reward will" is wrong, but some variation such as "lots of reward with a bit of punishment work much better than lots of punishment with a bit of reward" IS true, or is true for at least one of my kids. (But then, this weaker claim is something I already believe and it abandons the "never punish" enlightened part that was what really made it attractive to The Atlantic.)
Also, theories like these, even if not true, remind me not to overgeneralize from my own observations.
Their theory: Punishment doesn't change behavior. My observation: Yes, it does. So my theory: Punishment changes behavior. The truth: No, it doesn't. (As in, not necessarily) Me: Oops.
So, I'll keep reading and considering, but no, the basic claim of "don't punish, because it doesn't work" is wrong even if rewards also change behavior.