Supposing cows yield a better ratio of cut to ground meat (we've been breeding them for meat production and process them on an industrial scale), you'd still have to raise and kill a whole lot more cows to satisfy the world's current beef demand. I think good food safety practices and accepting a low, controlled risk of food-borne illness are justified by not having to kill as many animals.
[0] Not looking to start a discussion about the ethics of hunting. Not being in meat processing, this is an example I have first-hand knowledge of.
This is more a culinary and ethical argument than efficiency or safety. One should use the whole animal, in all its varieties, because they're delicious. Grinding otherwise edible parts of an animal killed on one's behalf because one disfavors (or is repulsed by) them seems to me much worse than the killing. Eat - and enjoy! - all of it or don't kill it.
It’s definitely better to freeze the left over bits and grind it yourself, but half the animal gets ground up regardless.
Who are you to say that it's wrong for me to enjoy some parts of the animal ground up, versus in their original shape? Either way the entire animal is being eaten and enjoyed, so there's no reasonable moral basis on which to say it's wrong. Your position would make sense if people were throwing away the cheek, tongue, etc but we aren't.