[Cushman]

Well, again, relativity doesn't violate causality. No one sees an effect before a cause. That translates into the cosmological principle in that all observers will see the big bang happen before anything else, all observers will see stars form before planets, et cetera, so it all comes out in the wash.

Nothing about that privileges a comoving observer with relation to unrelated events, though; they'll have their perspective just like everyone else, and that will be that. There's nothing "universal" about comoving time.