Yeah, killing during their play/hunt is the exact issue.
You claim dogs are way more of a risk, but our dog has 100% access to the chickens (without us being there) and protects our chickens from wild foxes. I trust him completely with the chickens.
I have a pet large munsterlander who's allowed access to the garden (with the chickens, about an acre) whenever he likes through a dog flap. I think he's one of the reasons we still have so many chickens, because of the foxes around here.
He kills pheasants (when he catches them, he's usually...