Time was when the .22lr was legal for use on roe deer in Scotland. In the right hands it was pretty effective, Henry Tegner shot many a roe with a semi automatic .22.
Equally, in the wrong hands, a lot of destruction can be caused with the larger calibres, for no discernible improvement in terms of lethality. I had a fondness for the 1oz Impax load in my shotgun, which I always found perfectly adequate for doing the job, ie delivering sufficient lethality to the quarry but without destroying the food value of the bird or rabbit; similarly, with a decent bullet (nearly always Norma’s 50grain soft point semi pointed) the .222 was and still is a great round for either fox or roe deer.