It would certainly expand on it, but it's actually not strictly necessary within DMC3's own plot. Vergil challenged Dante through Arkham, so he picked up the gauntlet to show him who's boss (things didn't go as he had hoped, but that's another story).
Yeah, but there isn't a clear explanation as to
why he's challenging him as a whole---outside of the whole world domination scheme involving Dante owning the other amulet---to explain their "sibling rivalry."
Lady mentions something about their constant fighting over the years having something to do with "twisted pleasure", but that hardly explains the amount of drama or tension the story tries to pent up with each confrontation they have.
I think a better way to establish their relationship would've been to depict their relationship children---maybe highlight the different choices they ended up making, and the different paths they chose after Eva's death.
What drove them apart?
What caused them to loathe each other like this? Did they have a brotherly bond before all of this? If so, what was the nature of it, besides the Jackpot joke referenced in the final confrontation with Arkham?
Anything's better than leaving it up to shonen logic of aesthetic differences, and riding on the paper-thin plot from there.