He said every deer he saw while hunting was being chased by coyotes.
I dunno--sounds like a bit of hyperbole on his part. I've never seen a coyote actively chasing a deer. A healthy adult deer is not normally going to lose a race with a coyote, and the yotes surely know that. Thus, the loss of piglets, goats, cats, dogs, etc. They gotta hit what can't get away easily.
I've watched them sneak around out West where I was muley hunting. Sometimes I'd see them from hundreds of yards away. Never saw them get after a muley and no real indication from the deer that they were harassing them. The mountain lions.....nother matter! But, very few of them and very solitary.