Barret rage and tantrums are supposed to be character flaws this time around rather than mostly comic relief. In one of the story changes
spoiler (click to show/hide)
where they save President Shinra but then he almost kill Barret in return
is pretty much because he is being an impulsive asshole that needs to make a show of everything and fucking Shinra calls him out that. Cat Sith did something similar in the OG game but much much later. Part of his character arc was that he realizes that he is a pretty bad leader figure and that Cloud, Tifa or even Cid are better in that position than him.
For better or worse, part of Barret character is that he is an attention whore. He is the guy that uses sunglasses at night.