Gawyn is not a citizen of Tar Valon. At the time when the letters were written. he wasn't employed in any way by the White Tower - he was not a part of the Tower Guard, the Younglings were getting disbanded, and he was not a Warder or even in training for a Warder.

And this is not a routine letter between lovers/spouses about business matters which can be left to the secretary. They've had a major fight, and Gawyn was so hurt he left the Tower. Yet Egwene was callous enough that she didn't take 5 minutes to write a warmer and more polite letter asking him to come back even though she knew Silviana disliked Gawyn and would probably rudely order him.