Birch dowels are my standard arrow material (after natural shoots). Birch is strong, stiff, relatively light in weight, and can be hand straightened (minor corrections). I've made several dozen from birch.
I make short, Native American style arrows, so when I buy a batch of dowels (mail order) I don't have too much waste (maybe 30%). If I am making arrows that require 30" long shafts, I expect only about 1 in 5 dowels to be usable (80% waste). Others might not have this much waste but I'm REALLY picky.....the grain has to run continuously from one end to the other.
I've found that birch is a little brittle for stump shooting.....with breaks occurring within the first 3" of the point. I break about one birch arrow per day when stump shooting (actually more like limestone chunk shooting around here).
As far as picking them up at the hardware store (and other places)....just sort through the dowels and picks the best ones. Most of these dowels will be Chinese birch and they work fine. (If they say "Made in Honduras" then they are probably not birch) They are a little lighter and more brittle than domestic birch.