It's purely personal preference, but it bugs me when I'm 'undercover' after I've done things that should've put me on the Thalmor's radar. In the Greymane liberation mission against the Thalmor no one who sees your face, knows of your involvement, and is on the side of the Thalmor lives to tell about it. How could she possibly not know who you are, especially if you're the Archmage? Elenwen would almost have to have known about if not given those orders. The Thalmor have received specific orders regarding you personally. The same 'anonymous officer' weirdness attaches to the College of Winterhold questline I'd say any time after the Dwemer mission. (If you're an Imperial officer you could always pretend you're leveraging your officer status to gain social rank it's a perfect cover.)

They would know who the officers are and not trust them. It really bothers me if I become a Stormcloak officer before taking that mission because it doesn't make sense for Delphine to tell you the Thalmor have no idea who you are when they very clearly should. There's a quest in the main line where you go undercover.