Erin, while you are 100% right when it comes to most things, to get a dog to consistently leave something on command you generally need to offer a higher value reward than the thing it's trying to steal. My Jack is trained through TDI certification. I can leave food on the coffee table and walk him off-leash past treats on the ground. He has an outstanding recall and a fabulous leave it.
But nothing I can possibly give him is tastier to him than cat poo, and I have never been able to get him to leave cat poo if he finds it buried in the leaves or something, not even if I am carrying steak or cheese or liver treats.
If what you are trying to get the dog to "leave" is the single most rewarding thing in the world to the dog, and it's a smart dog, it will not come off it no matter what. The dog understands that what you are offering is not as good as what it is after.
Aversive methods (positive punishment) have their place. A vibrating collar won't hurt the dog. You do have to be very careful when using them though and they are not for everyone. I used a squirt gun to the face when Maddie was stalking the cat because even waving food in front of her would not break her concentration. It only took a single squirt on two different occasions to get her to realize that she needed to look at me when I said her name, no matter how fascinating the cat was, and THEN she got her reward.