Personal opinions, some relevant to the OP, plus more (at risk of being in the Department of Redundancy Department) :
In my experience, wood type, combined with craftsmanship, matters more in its inherent ability to maintain shape (true round cylinder, true flat [except for snare beds of course] and consistent bearing edges) than as a tonewood. Shell integrity variation is what produces varying ability to enable the heads to do what they're supposed to do.
The heads are much louder than any sound produced by the shell itself. That said, with all other things equal (size, bearing edge shape, tuning, etc) I feel I can hear more difference between hardwoods and softer woods, than I can between types of each- i.e. I'd be hard pressed to tell the difference between hard rock maple & bubinga or poplar & luan, but I can hear a bit of difference between one of the first pair and one of the second. With toms and with snare drums with the snares off, the tone difference is faint; with the snares on, it's even more difficult. Metal shells make a more distinct... well,
metallic ping sound so it's easier to hear their tones' differences from wood. However, I once owned a Legend 5.5 or 6 x14 rock maple snare that, with its die cast hoops and flawless craftsmanship- including very consistent 45° edges- sounded more like a metal drum than any other wood snare drum I've owned- to my ear, indistinguishable with the snares on.
I can't speak much to the tone of birch, as I have very little experience with them. While working at an instrument store nearly 2 decades ago, I was very underwhelmed by a Tama Starclassic Performer snare (pre-blended wood, all birch) that sounded boxy and dull no matter the tuning of either head. (That stung, because I was and am a Tama fan.)
This very early 1st gen Export birch snare drum sounds pretty good so far, but I haven't put it through its paces after putting a better head combo on it.