Birch is not strong enough nor stable enough to use for stocks, generally. I suspect from your location, you are looking at paper birch, which is the worst of the birches. Yellow birch is far better than paper, especially in stability.
For a rifle of modest recoil it is possible to do.
Drying birch is difficult because it dries quickly but not uniformly which leads to warping issues. Air drying is difficult to control. Most stock makers refuse to use kiln dried wood.
While birch may have plenty of figure it has very little "depth."
