I think if you want to make shooting crows easy but expensive go .17hmr, don't bother with the .22 mag. But if you want to become a better shooter, and have to employ a better understanding of ballistics to shoot the crow, go .22lr. There is no substitute for the .22lr when it comes to sheer practice time, low cost, versatility of ammo selection etc... Another thing to consider is that many .22lr firearms will shoot hyper velocity ammo quite well. A stinger going 1600+ fps has close to the same short range energy as a .22 mag. Even velocitors have an impressive 1400+ fps velocity and they hit really hard. So really how much do you want/need an extra couple hundred fps or extra dozen ft/lbs of energy? It's really not a big difference when were talking about crows under 100yds. Plus you get the added versatility of being able to also use subsonic ammo for discreet plinking or small game hunting without attracting a whole lot of attention.
.22lr is superior for those reasons in my book. If you really want more than that, the next logical step up in my mind is .17hmr, it's very fast and flat shooting so it's easier to hit targets far away because you don't have to factor in drop, but it blows around in the wind easy, it's really loud for a rimfire, and it gets really expensive to shoot.
Just my 2 cents.