Why didn't Taki write his name on Mitsuha's hand?
I think it was a sweet gesture that showed a little insight into Taki's character. It shows that he is young, in love, and struggling to find a way to share his feelings with Mitsuha.

At that moment when an opportunity presented itself for him to confess without having to say it out loud (the horror! :D), he foolishly thought it was more important to tell her how he feels than what his name was.

Very frustrating, probably done on purpose by the writer but not completely baseless.