Positive rewards & problem solving

I know people who don’t believe in positive reward training. There’s lots of you out there — you who believe that the best way to teach a dog to behave is just to punish it when it does something you don’t like.
I disagree.
Using positive reward training teaches your dog to problem solve and gives them an outlet to do so.
Here’s the scenario:
You have a treat. Dog is looking at you. “What does she want?” the dog wonders. You say lay down. The dog sits. You say lay down. The dog whines. You say lay down. The dog sighs. You say lay down and point at the ground. The dog looks at the ground, whines again. You say lay down. The dog gets bored with this game and lays down.
You give the dog a treat, and now the dog is thrilled — happy and exuberant and ready to do it again.
With every positive reward exercise, your dog is learning to exhibit different behaviors until he gets the one that is rewarded. It’s a process of elimination for him, and it’s problem solving.
Sitting didn’t work, so I whined. Whining didn’t work, so I sighed. But when I laid down, I got the treat. Problem solved.
Would you rather your dog apply his problem-solving abilities to trying to figure out what you want from him, or trying to figure out how to turn door knobs or open garbage can lids?
Take your pick, because his problem-solving skills are going to come out in some form.
I prefer the one that makes life easier for both me and the dog.