I have killed several antelope with 22-250s. I started out with 80gr Amax which worked really well with a chest shot, I shot one for cross eyed with it which knocked her out but didn't make it to the brain and she took another round in the heart. I switched to 62gr Barnes TTSX at alot higher velocity. I shot one the very next year at almost exactly the same distance and placement, the barnes blew out the back of her skull and zipped across the desert.
I have no problem killing things with 22 calibers but I wouldn't let my kids use them on their first hunts.
I also shoot alot of my big game animals in the head. I don't suggest it but I do it pretty often.