I agree with you, sort of. It at least isn't clear that good would come of it.
When it comes to calling something a crime, I don't give someone a pass because they're the government-- locking someone in a cage because you don't like the drug they used is the act of kidnap and that's a crime. But I don't hold them to a stricter standard either, and sending emails in a certain manner isn't actually "a crime" when you just pretend you're an alien who doesn't have the context of her job title.
It's similar to like a terms of service violation, it's a violation of US government protocol that you agree to when you become a member of it.
A violation that could have huge consequences, sure. But when you get over it being technically a crime, you just have to be ok with however the government system chooses to police its rules (just like you have to be ok with what YouTube thinks isn't allowed on its site). So I don't see any inherent need to prosecute her as you'd need to if it was an actual crime like rape or assault.
"Hanging out our dirty laundry for the whole world to see not only makes us as a country look bad, but vilifies our elected officials"
I think time-preference has a bit to do with it. Hanging out the laundry seems certainly more bad than good short-term, but it may lessen the chance of it in the future.
I don't lose any sleep over what they decide, because it would just be playing wack-a-mole anyways. It's a natural function of democracy that the most corruptible people will compete the hardest for the positions of power and over time win them. So even if it's "bad" in a sense, it's on the other hand good to not pretend there's actually a way to stop a democracy from becoming corrupted.