Walnut has always been the best wood for most gunstocks.
For pistol grips, it really doesn't matter very much. The grip is just a small handle beween the steel and the hand. It doesn't get stressed very much. The cross grain section is not very large and the shock of recoil isn't going to damage it much.
Long guns on the other hand need a long thin piece of wood to take up the shock of recoil. A long thin piece will be much more susceptable to bending stress. Walnut is ideal. It is strong. it is not as hard as oak or maple so it is easy to shape with hand tools. Ash is an ideal resiliant wood. That's why it is use for baseball bats. But ash, like oak, has a strong open grain and is suceptable to splitting. I have never heard of a gunstock being made of ash.
Maple was fairly common with craftsman type gunsmiths building rifles one at a time, lock, stock, and barrel. Highly figured maple was prized for its beauty.
But once mass production came along, with template following machines to mass produce gunstocks, walnut pretty much became the standard. Sycamore is pretty similar to maple in it's properties. I've never heard of sycamore being used for gunstocks.
Frankly, I'm a little bit surprised to hear the bit about birch. Birch is pretty much viewed as a second rate wood in the hardwood world. It's properties are pretty similar to maple. Quite hard and durable. It seldom has any exciting figure at all. Birch was often cut into 4 foot long bolts and used to mass produce small wooden parts for industry. Shoe lasts for the shoe industry were usually made of birch. Clothespins too. Come to think of it, if you needed a no nonsense, durable, if a little bit boring wood, birch was probably a good choice for gunstocks.