The most interesting part of this is that Goku is actually a "good guy". He's just overdoing it with the fighting. I've seen people say he "deliberately" wanted to destroy the universes, but that's never the case.
So, I'm thinking what a DB-like character arc for him would be like. Keep it simple, but do something.
A) Vegeta/Piccolo gives him a tongue lashing to make him understand the severity of the situation. Vegeta in particular would really be effective because he's expecting a new child! "What if YOU were having one, Kakarot?!"
B) a universe loses and gets destroyed, just when Goku has come to enjoy fighting someone from there….hits home? Slowly but surely the consequences become more of a concern. "Wait, OUR universe could go next?!" probably when half the team has been knocked out or something.
C) Toei: "lol develop Goku? nice one"
And this is more of a tinfoil theory, but I do not trust the Grand Priest at all. Sure Zeno is child-like and doesn't articulate things well, but his mouthpiece is coming in with all of the spoons to stir the pot -> "oh Goku-san brought this tournament up" "btw Zeno-sama was going to destroy anyways" "U7 and U9 are ranked the lowest, tee hee!" "the tournament has rules unlike this exhibition match"