As a man who personally loves the 10/22, but has shot grouse and seen them fly away into the dense brush with a bullet in them....
I only use my 20 gauge coach gun.
So basically what I'm saying is.... small game that you can take with a .22lr, you can take more reliably with a shotgun. You know the saying "A bird in the hand, is worth 2 in the bush."
And don't be affraid of blowing them in half with it.
I've taught lot's of new guys how to shoot grouse and rabbits with a 12 gauge, and they literally all took my advice and shot their heads off, and didn't waste an ounce of meat.
Basically... with a shotgun, you have a much wider range of targets. i.e flying, running, standing...
With a .22lr..... you have to get super lucky to find one standing perfectly still.

Or... best small game gun ever?? Stevens Model 24.
It's an over/under rifle-shotgun. You can still get them today in a .22lr/20 ga.
Check 'em out.
http://www.gunblast.com/Savage-42.htm