The disclaimer must also state that the dietary supplement product is not intended to "diagnose, treat, cure or prevent any disease," because only a drug can legally make such a claim.
No; they can make health claims, but they must be supported by scientific evidence. And there is tons of research for them to pull from — I only sent you links to a subset of English language studies — and they can fund their own studies.
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https://www.fda.gov/food/nutrition-food-labeling-and-critical-foods/label-claims-conventional-foods-and-dietary-supplements#:~:text=If%20a%20dietary%20supplement%20label,legally%20make%20such%20a%20claim.