The author distinguishes between the need to act hastily to stop a behavior (to protect the cat, in your example, or the child from himself, in the article's example), and what we do as a follow up (time-in spent talking and engaging instead of time-out spent isolating).
As we've known for a long time, positive reinforcement causes people (and animals) to repeat behaviors that resulted in being rewarded. Negative reinforcement, on the other hand, can sometimes stop unwanted behavior, but it can just as easily cause people to become better at hiding it, which is a lose-lose. First there's the loss of trust at becoming the person the child has to fear, and then there's the loss of connection by not really knowing what the child is doing. I'm sure many of us have had "if my parents knew..." moments. Maybe we still do! We hide things from people who punish us. (It's a common problem in adult relationships as well. If our S.O. punishes us for something they don't like, usually by withholding themselves, then we start to hide that part of ourselves from them, and vice versa. Rarely do we change our behavior because we were punished.)
Anyway, I tend to agree that we should use positive reinforcement whenever possible, but it does require a lot more time and energy than negative reinforcement. Punishment is much easier to dole out than finding effective rewards. With dogs, you can reward them with a piece of ham, and it will never get old, ever. With kids, yesterday's reward is today's tedium, but punishment doesn't require much creativity or reinventing (unless you enjoy that sort of thing). And negative reinforcement may not be as good, but it would be a lie to say that it's completely ineffective. And some behaviors are only inappropriate in public anyway, like picking your nose, so it doesn't matter if it's hidden. Negative reinforcement can accomplish that, so it's just a question of whether it's worth the weakened bond.