I think Annatar should be less relatable. He should be pretending to be an elf or elf-adjacent being, but in way over his head. He didn't think this through all the way. Centuries of living with Morgoth and the orc gang left huge gaps in both his general knowledge and his understanding of socially acceptable behaviour. But at the same time, he is fantastically overconfident in his own abilities to the point where everybody just assumes that he always knows what's going on even when he clearly doesn't.

Like Celebrimbor says, "Hey Annatar, pass me that fork, will you?" And Annatar, having spent the last few millennia living in a literal cave and having no idea what this weird elven eccentricity called a fork is, says, "The what?" And then Celebrimbor just laughs, thinking silly Annatar is joking again, and grabs the fork for himself.

Nobody ever catches on.