"Yes, whatever the duration, you'll eventually stop vomiting some time after taking the medicine."

That's a gross oversimplification. You could continue to vomit. You could also die. There are multiple types/causes for vomiting. Not all vomiting simply stops on it's own.

"It's good old mind-tricks of correlation vs causation."

I'd say this is more than just mind tricks. The study shows there's a strong correlation and evident effects, which suggests studies should be done on the cause.

"There's a reason real medicine requires preregistered double blind trials."

Or in this case a grandfathered status (it is a "real" medicine as it is approved for sale). I also think this is, again, only a half truth. Double blind studies tend to provide the strongest evidence, but they are not the only kind of study accepted for drug approval.