Two reasons. First, using a stick as an item or a weapon is a child-like idea, same with the Boomerang (real boomerangs are actually toys) and Slingshot (same), so they simply made all child-like weapons only available to Young Link.
The second is the unpopular "it's a video game" sort. Basically, it's for design reasons. The Slingshot is replaced by the Bow. The Boomerang is essentially upgraded to the Hookshot (which, at least against enemies, does the same thing: stun them). And the Deku Sticks are basically removed to make the torch puzzles (which are also a little child-like if you think about it) more complex, or in some cases remove them entirely. The only torch puzzle I can recall after the Master Sword is acquired, actually, is in the Forest Temple where you have to shoot an arrow through a torch to melt some ice.
Deku Link drops Deku Nuts, but only while flying and they actually consume his Deku Nut stock. I also think the idea of spit bubbles or loogies done like that are a bit of a stylized thing over in Japan. While I might be mistaken, I'm pretty sure of that, so maybe that's one reason for it. They also probably wanted to give him a more harmless attack since he was only a Deku.