Humor and despair
I've not read one word of Peterson's work, and I have no desire to defend his ideas per se from this charge. But the critique, though plausible and even probably true as far as it goes, fails in two respects. And it does so whatever the quality of Peterson's thought in itself.
In summary form: the critique fails, on the one hand, because it presupposes a priori that despair is not a viable or rational option for considered reflection; and, on the other hand, because it assumes a modestly affluent and pleasurable quality of life sufficient to ground the humor that winks or smirks at, or spits in the eye of, the abyss of death.
In other words, Burkeman (the author of the review) recommends a coping strategy at the practical level regardless of its truth at the theoretical or existential level. To which the Petersons of the world are justified in replying: I don't want to cope with the un-copable. I want to live in the truth. And if the truth is the sheer implacable terror of mortality, of the inexorable power of unconquerable death, which will swallow up me and everything and everyone I love, rendering us not only nonexistent but our lives and loves meaningless—if that is the truth, then I would rather suffer that terror humanely, truthfully, and therefore humorlessly, than play-act with jokes and empty grins.
The other side of such a reply is that one can only imagine coping with finitude through humor no matter the truth of the matter if one's material conditions rise above a certain level. That is, if my day-to-day activities include modest pleasures and even delights—fulfilling work, faithful marriage, beloved children, delectable food, enjoyable leisure—then these can provide respite from the existential torment of endlessly meditating on suffering, death, and loss. And in such respite one is permitted the therapy of humor, even gallows humor.
But again, here a Peterson is able to mount a reply: Our affluence may be nothing more than a conspiracy to hide the truth from ourselves, indeed society itself may be just that: a systematic attempt to make us forget, for sustained stretches of time, the absurdity of our lives and the single common fate awaiting us all. And, again, if the highest aim of the human creature is to live in the truth, then better to resist this massive operation at collective deception; better to live, miserably and humorlessly, in the truth that quotidian pleasures try to mask, than to live a falsehood that includes laughter.
Put differently, if humor is nothing but a way to cope with suffering, then it is perfectly reasonable to decline the invitation, given a different hierarchy of values. But if it is meant to be more than merely a coping mechanism, then more is required to ground it in the soil of the real. It must arise from and share in the truth. If it does not, then despair may be a legitimate response to the realities of bare human life (and nothing is less funny than despair). At the very least, it cannot be ruled out in advance.