It's not exactly even that. After the segment changes between the second and third set of dates and the major conflict drivers of the game are introduced, Helltaker is suddenly aware of what is going on (and who to watch out for) with little explanation of when or where this was explained to him. I understand that characters like Lucifer are more aware, but I feel like another scene or even a brief interaction would explain what has set this up.
Ok, that aside, once more, this is a legitimately fun game and I adore what is being done here. I wouldn't be surprised if my little gripe is unfounded and I've skimmed over something.