Textbook (Copenhagen) formulations of quantum mechanics are inadequate for cosmology for at least four reasons: 1) They predict the outcomes of measurements made by observers. But in the very early universe no measurements were being made and no observers were around to make them. 2) Observers were outside of the system being measured. But we are interested in a theory of the whole universe where everything, including observers, are inside. 3) Copenhagen quantum mechanics could not retrodict the past. But retrodicting the past to understand how the universe began is the main task of cosmology. 4) Copenhagen quantum mechanics required a fixed classical spacetime geometry not least to give meaning to the time in the Schrödinger equation. But in the very early universe spacetime is fluctuating quantum mechanically (quantum gravity) and without definite value.There is some merit to this reasoning, but jumping to Everett many-worlds is still bizarre, and does not help.
The decoherence and consistent histories interpretations of quantum mechanics are really just minor variations of Copenhagen.
While Copenhagen says that observers notice quantum states settling into eigenstates, these newer interpretations say it can happen before the observer notices.
Many-worlds just says that anything can happen, and it is completely useless for cosmology.
Sean M. Carroll has announced that he is writing a new book on many-worlds theory. He will presumably take the position that it is a logical necessity for cosmology. Or that it is simpler for cosmology. However, I very much doubt that any benefit for cosmology can be found.