I believe they shouldn't.
When I played through Three Houses for the first time recently, I thought to myself "Why could I rename Byleth but not actually customize his appearance?". I think interactions with your students and the other characters would feel a lot more personal if the protagonist had a set name, instead of constantly being called Professor. If you're going to have the option of changing the protagonists name, you minus well go all the way and let the player customize the appearance as well. Only having the option of changing their name just seems pointless to me.
(Just realized a similar post was made last month, good job me.)