Smaller calibers are fine. I used to do all my own butchering and have dropped countless beef with a 22LR. Shot placement is what counts.
As a kid I was given 3 shells and told to come back with 3 meat animals. I learned real quick. Learn to focus and take head shots even at deer.
My neighbor here in rural MO says more deer have been taken here with a 22LR than anything else. Shots often taken out the window of a truck and yes, head shots.