Now that I think of it further, I wonder how Bethesda themselves caused dates to appear in all the earlier games? Because when a quest gets completed and moves into the Quests Completed section. This usually happens due to a SetStage command or some sort of variable being set, which are visible I(and changeable, for those who like playing risks) n the Construction Set.
But what's missing is any thing which sets the actual date. There is no SetDateFinished or anything like this. Which makes me think Beth hardcoded the game this way, meaning we don't have immediate access to whatever they did to make dates appear.
So whatever they did in the earlier games, they did not bother with for Skyrim, for whatever reason.