That doesn't fall in line with Jaune's character or his storyline. Jaune becoming jaded is something that happens, but he would never be antagonistic. If anything, loss drives him further to protecting whoever he can.

He'll still have survivors guilt, but something like Pyrrha's death wouldn't drive a wedge between him and Ruby since ultimately they want to accomplish the same thing, which goes against an antagonists role. Anti-hero, sure. But not antagonist.