There's a reason you don't see many birch stocks au natural. Generally speaking, birch isn't known for it's stunning grain figure. Depending on how the wood is cut, ie 1/4 sawn, flat sawn, etc. The grain can run from delicately pretty to basically a pine 2x4. Pre-conditioner is usually a good idea for something like birch paneling or plywood but comes at a price of reducing the ability of the stain to penetrate.
Wood stains like Minwax are known as pigment stains. They are micropulverized pigments with a bit of dye attached to the binder. What you are stirring up from the bottom of the can is pigment. It works by getting into the pores of the grain and then gets sealed there by the binder and any finish you decide to put on. I've found that in any wood where there isn't much grain to begin with, using a pigment stain usually leaves it muddy looking when you look closely at the wood, regardless of the shine you might put on with the topcoat.
My advice would be to use a dye like Transtint. If your piece has nice grain structure you would like to highlight you can dye it black and then sand off the surface leaving only the black hidden in the grain. Then dye it with whatever color you choose. Or if you want to get fancy, you can layer different tones to create some warmth and depth to the color.
Pay close attention to your sanding direction. Use a sanding block wherever possible. Your thumb will dig into the soft wood faster than the hard part, especially if the wood is flat sawn. The dye doesn't know the difference between natural grain and a big ol' cross grain sanding scratch either.
Your best bet would be to find a hunk of birch and practice on it before trying it out on your gun.
