I started in with a .22 in 1940. '06 in 1950. I've done in a fair number of critters through the decades.
"Knockdown" is not something I've ever really worried about. Empirically, a .22 is fine on little stuff. Centerfire for bigger stuff. Great big centerfire for dangerous game.
If you break the shoulders on a four-legged critter, it can't go anywhere at a rate of speed which would endanger the shooter.
Medico-scientists have stated that heart shots on the monkey-clan (which includes people) stops/kills faster than on, say, deer. It apparently is a physiology/adrenalin thing.
All of which is a circuitous way of getting back to shot placement.
You pop Bambi in the white spot and he's through for the day. .222 through '06, I've noticed.