Lots of things that are completely useless have been used for thousands of years for their medicinal purposes. The evolution argument only holds if there were some negative effects here. Anything that doesn't do harm to the body can easily integrate into the huge body of knowledge called superstition. Eating it is tasty and doesn't hurt, there's nothing more needed to explain that Asian cultures have it as a medicinal herb.
If it for example was good for you but caused somewhat unpleasant side-effects, your evolutionary argument might have more merit. Anything that works as well as homeopathy (it doesn't), but doesn't harm either (like homeopathy), won't be sorted out.
That is utter nonsense. To give just one counter-example, bloodletting was used for millennia for conditions where it’s completely ineffective (and, in fact, actively harmful: bloodletting has killed scores of people). In fact, you simply can’t apply an argument from evolution here. Evolution doesn’t optimise for “goodness”. It optimises something (and only over the long term). That “something” can often be positively detrimental to other outcomes, and even to general fitness (e.g. the laryngeal nerve).
These assumptions are more relevant in small sample sizes. Making them inherently unreliable. What's missing is not a p-value, but an estimate for the uncertainty around that p-value.
Statistically, there is no such thing as a 'small sample size', in isolation. It is always related to the power and effect that the researchers want to work with.
Moreover, even if a study is statistically insignificant, it DOES NOT mean that the opposite conclusion is true. It might often merit some further examination of cultural and historical trends. In this case, for instance, turmeric has been used for medicinal properties in Asian cultures for centuries. Theories of evolution suggest that this wouldn't be the case (and it would've died a natural death), if there wasn't some merit.