As suggested in a comment, this question may not be objectively answerable, but for what it is worth, here is a quote from the book What is Real? by Adam Becker (mentioned in another comment). Becker proposes the following narrative, which argues (as mentioned in yet another comment) that John von Neumann played an important role.
Yet Einstein’s pleas for a more complete theory went unheard, in part because of John von Neumann’s proof that no such theory was possible. Von Neumann was arguably the greatest mathematical genius alive. … His colleagues at Princeton said, only half-joking, that von Neumann could prove anything—and anything he proved was correct.
Von Neumann published his proof as part of his textbook on quantum physics in 1932. There’s no evidence that Einstein was even aware of this proof, but many other physicists were—and for them, merely the idea of a proof from the mighty von Neumann was enough to settle the debate. The philosopher Paul Feyerabend experienced this firsthand after attending a public talk given by Bohr: “At the end of the lecture [Bohr] left, and the discussion proceeded without him. Some speakers attacked his qualitative arguments—there seemed to be lots of loopholes. The Bohrians did not clarify the arguments; they mentioned the alleged proof by von Neumann and that settled the matter…like magic, the mere name of ‘von Neumann’ and the mere word ‘proof’ silenced the objectors.”
At least one person did notice a problem with von Neumann’s proof shortly after it was published. Grete Hermann, a German mathematician and philosopher, published a paper in 1935 criticizing von Neumann’s proof. Hermann pointed out that von Neumann failed to justify a crucial step, and thus the whole proof was flawed. But nobody listened to her, partly because she was an outsider to the physics community—and partly because she was a woman.
Despite the flaw in von Neumann’s proof, the Copenhagen interpretation remained totally dominant. Einstein was painted as an old man out of touch with the rest of the world, and questioning the Copenhagen interpretation became tantamount to questioning the massive success of quantum physics itself. And so quantum physics continued for the next twenty years, piling success upon success, without any further questions about the hole at its heart.
As an aside, I do not think that the Copenhagen interpretation “reigns supreme” nowadays. Textbooks that spend any time discussing the philosophy of quantum mechanics typically mention that the topic is controversial, and briefly describe several different approaches.