While such a figure is RAW, I would note that
according to Kromm, such a person would qualify as "mythic" in skill. Maybe not Asclepius, but certainly his sons (Machaon and Podeleirios).
No, I would not allow such specialization in my games. It's just a boring character, a walking band aid kit. I suppose you could add disadvantages and quirks to make him interesting, but... it just doesn't appeal to me as a player or as a GM.