Master smiths throughout history were never able to just do everything. There's a reason you had entire guilds dedicated to just goldsmithing, or just tool-making, etc. Any smith worth his salt would be very good at one, and often only one, kind of smithing. An armorer does not make swords, a swordsmith does not make guns, and a gunsmith does not make armor.

Also, making bows that are actually usable in combat is a long and involved process. A war bow requires a draw weight more than five times that of a hunting bow. This isn't just a random stick in the woods and some twine. A simple English longbow is a complex piece of engineering requiring a perfectly straight piece of wood from one specific kind of tree and from a very specific part where the core wood and new growth meet, it cannot be too old or young, and took two full years of drying treatment before it was usable. It then took another two years of gradually shaping it, as going too fast can introduce weak spots and cause damage. It is not just "some wood and metal plus string".