I am guessing in this instance it's a dramatic tool to coldly inform Ennis of Jack's passing by the filmmaker, with the presumption this would be a normal procedure (it's been used in other movies and TV shows). Most mail carriers still deliver mail for those no longer with us, or if they lived alone, it gets returned as undeliverable. We are unfortunately still dealing with this in our own family as my father is closing charge accounts opened in my mom's name (they all want a copy of the death certificate -and- mail a confirmation letter in her name at her address because a third party is closing the account). And he still gets junk mail in her name as well.
I don't sense Lureen would be -that- cold as to wash Jack right out of her scary hair, but since a lot of us think of her as one of the things which keeps Jack and Ennis apart, it's probably easy for us to assign negative intent to her. I've noticed Alma gets far more sympathy than Lureen does in a lot of discussions about the women. I wonder if that is part of the stereotyping of strong, take charge women as cold, while Alma clearly looks more like a victim.
One wonders how Ennis would share the news of Jack's passing with Alma.....