The puzzle rests on translating tanha as “desire.” But tanha is craving or thirst, and the second Noble Truth never said all desire is the problem. Isn’t the “concession” just the doctrine correctly understood?

Mostly yes, and the word concession is wrong; the distinction between craving and wholesome desire is original to the tradition, not something it was caught out by. The puzzle is a puzzle for the popular slogan Buddhism says desire causes suffering, not for the doctrine. What survives is smaller and more interesting: if the tradition distinguishes good desire from craving by the object and manner of desiring, then the deepest problem is not desire itself but misdirected desire, which is close to Augustine’s analysis of disordered love. That is a point of contact rather than a defect.