Yet a spear isn't? Honestly, even after the den, you likely won't have enough to afford a gun; and I would venture to say most wouldn't find the gun hidden in the rat's layer, soooo...why not?
Besides, a bow is good for RP purposes. The game has throwing knives, arguably the most useless weapon in the game next to sharpened sticks and rocks. So why not a bow, even after leaving Arroyo?