Reality swirled in a wholly blue manner, revealing absolutely nothing of merit. Things flashed madly in and out of existence like some entirely other sort of phenomena. On a whim, Kurt Vonnegut imploded, taking a sizeable hunk of Massachusetts with him. However, seeing as Massachusetts wasn't entirely sure it existed, the chunk wasn't all that big.
The whole "find a study" argument is, to be blunt, complete crap. Why? Because for it to be done, you'd have to take a load of new-born babies, and carefully monitor the way they grow up. Some in isolation, others in different countries, others in different social situations, different types of school, some with corporal punishment others without and not to mention the repeat experiments; all of this just to, as you mentioned, isolate the variables to be sure whether it is the corporal punishment (or lack of) having and effect.
The whole thing would be infeasible.
All of us have experience of either corporal punishment, or not, and all of the different variables, so we must pool our own experience to act instead of a study.
Personally, I think that corporal punishment is not always right and not always wrong. Schools SHOULD NOT DO IT. I know of many malicious teachers in my history who would be very dangerous to be allowed to be in a position to hurt children.
With parents it depends on the circumstances. It's all about proportion, and reason. The child has to know what they've done wrong and why it's wrong or there'll be no point to it.
Even if such a study were unfeasible, the fact remains that the burden of proof is on the person who says that corporal punishment is effective.