I've never had problem killing squirrels with body shots with a 22 lr, and in my short stint in the butcher shop, indeed the steers went down when hit in the head with the old 22 they had. The big bulls, however, took the 30-30, because sometimes the big boys skulls were too tough for a clean kill. I always shotgunned or centerfire rifled the bigger pests, because head shots with 22 are the only humane option.
As for aggressive woodchucks, i spear them. If you catch them in the open, they don't run, so I fight them. Also have caught them in the big old polebarn where there is too much expensive equipment to damage to use a gun, so my spear, just a big old 15 pound iron crowbar, will do the killing well. One time I impaled one with a wooden spear, a big ole sow, and she had enough strength to turn herself around 180 degrees whilst being impaled to face me. They are not cute, nor cuddly.
As has been stated, heard twice of someone hitting a skunk with a .223 and the skunk being able to run for a bit before death. Saw the damage upclose and was disappointed. Far inferior to the 220 Swift. Sure, the animals went a long way after you hit them, but that was many pieces of them in many directions.
So, to get to the OP's question, 22 lr is a weapon that demands effective shot placement, it works, but you have to work it. You want something that doesn't fail as often due to poorer shot placement, get a centerfire rifle, high power pistol carbine, or good old shotgun. I love my old 22's a lot, killed a lot of things with them, but over time I've come to use shotgun for serious pest work, highly effective and humane.
I have the old Winchester 1906 pump action 22 our family used for slaughter, called the "pig gun", which has done great work and been used to kill countless pests in the yard. But then again, the time when it was built bought, and used, was different; the old farmers and men shot 22 long rifle and short rifle because it was cheap, and things were expensive, and you lived conservatively. Today, many more of us can certainly afford a more expensive weapon with more expensive ammunition for practical purposes.