They seem to be two separate "occupations" though. Death is basically subordinate to the Fates. He doesn't decide who dies, he just makes sure that those who do, well, do. When Balthazar went back and changed history, Death wasn't responsible for killing those people since their fates had been altered, albeit without the Fates' consent. Which is what pissed them off.
Death really had no involvement in these events since Sam and Dean didn't actually stop anyone from dying. Fate were just running about "course correcting" the events, and Death was coming in to clean up after her (which we didn't see because it didn't play a relevant part in the episode).