Powlison bases a lot of his critique on the fact that CBT uses (sometimes consciously) methods that can rightly be described as Stoic in that they do have a strong enough similiarity to key ideas of the ancient Stoics that I don't think the comparison is inapt. Stoicism, at least on the issues relevant here, involves one key claim. The Stoics didn't think it's worth worrying about something outside your control. The reason is that your life is made worse off by your worrying, but you can do something about the worry. You can't do anything about the fact that George W. Bush won the presidential election in 2004 or Barack Obama won the presidential election in 2008. You can't change the fact that lots of people died recently in China from landslides. You can do something to help those who remain, and you can do something to change people's minds on policy issues and perhaps help elect a different sort of person next time, but there's no point in worrying about something you can't do anything about.It's an interesting analysis, but I know too little about psychology to have much to add. I do know a little about Stoicism, although not enough to correct a professional philosopher. As I understand it, at its core, Stoicism was fatalist. I don't mean that in a necessarily negative sense, but in the technical sense. Stoics believed that the future was set, that it had indeed happened before and would happen again, in an endless repeating cycle. This didn't mean that anything you did was meaningless. You, too, had a part in that cycle, and what you did had meaning and relevance. But you had done it before and would do it again.
That element of Stoic philosophy seems entirely reasonable to me. The Stoics do go on to say that we should remove all emotions, but it's important to be clear on what they meant. They defined emotions more or less as bad reasoning. Things we call feelings that aren't bad reasoning and are compatible with good reasoning would not be emotions for the Stoic. So there's no reason to complain about that view on the ground that it's healthy to have emotions and inhuman not to. We should eschew the things they called emotions without actually eschewing emotions as we understand the term. They had a strange view about what we should call emotions, but the substance of their view is mostly right, as Augustine so deftly argued in his critique of the Stoics. Feelings of any sort should be submitted to reason, and those that are irrational are best removed. Augustine shows that the Stoic view, when reworked into ordinary language without their odd view of what counts as an emotion, is largely correct and fully compatible with Christian teaching.
Where the Stoic goes wrong, as far as Christianity is concerned, is in not submitting things to the lordship of Christ. I can't even say that they don't equate submission to reason with submission to God. They do. They just have a false view of what God is like. Does that affect the practical level? Not so much. Does it affect CBT? Not remotely. The reason is that CBT is really a method, a placeholder in which you insert the content you intend to replace the unhealthy and irrational beliefs. The Stoics insisted that irrationality comes from false thinking. They may have been wrong about that as a fully adequate explanation of all irrationality. But they were certainly right that a whole lot of irrationality comes from false beliefs. I know at least two cases of chronic depression that in large part involves flat-out false beliefs, even if there may also be neurological causes. In one case it's someone who consistently interprets any possible information that could be stretched to show that people don't like him or that he's a failure as if everyone doesn't like him and as if his abilities are the problem, when in many of these cases no one is even evaluating him negatively, and often enough their evaluations aren't seen that way by the people doing the evaluating. Such a person might benefit from neurochemical supplements, but CBT would encourage him to replace those false beliefs with a more hesitant approach to such negative interpretations, one much more like how most people would respond.
CBT is offered as a correction to the biggest problem [of] Applied Behavior Analysis therapy. ABA insists on treating only behavior without dealing with anything internal, e.g. unhealthy beliefs. It stems from the behaviorist model of psychology, according to which we shouldn't postulate anything internal that can't be measured empirically, and thus any psychologist who talks about beliefs, desires, and so on is engaging in unscientific behavior (notice that even the way I've constructed that sentence admits only to the behavior of such a psychologist; a behaviorist shouldn't even say that such a psychologist has false beliefs about how psychology should be done, just that the speech and methods of such a psychologist are unscientific).
At least, that's how I understand their beliefs.