It would kind of fit in with their sort of "futuristic yet not exactly" aesthetic.
Yes thank you ensign for writing that report up on your tablet computer, now because email apparently doesn't exist anymore you'll have to physically leave the tablet with me and go get a new one.
I know that this is a lighthearted place, but why do you think so? E-Mail is one of the most democratic and equity promoting forms of communication there is through its decentralisation and open standardisation. It would be the very kind of thing the Federation would want to preserve or introduce itself.
Email is incredibly insecure. There can be no expectation of veracity or privacy within an email. The core of Email protocols were made when the Internet was a much more secret club.