Team Fred and Team Sanity are BOTH RIGHT.
Paul and Casey were right in pointing out that the movie presented Fred as an external entity who exists independently of Elizabeth. But the movie is from the perspective of Elizabeth. And from her subjective perspective, Fred is an external entity who, at the end of their journey, would go and befriend other troubled girls.
From an objective perspective, though, June and Jason are right. It's simply true that imaginary friends are the projection of the individual's mind, regardless of whether that individual perceives it that way.
The irony is that June and Jason were accusing Paul and Casey (and all of Team Sanity) of losing their view of the world through a child's eyes, but in fact it is Team Sanity who sees the movie through Elizabeth's eyes, and June and Jason who see the movie through the detached, objective lens of reality.