Unless you're talking about trials with millions of participants, what you'll actually get is just noise.

And probably a lot of death or maiming from drug interactions.

Which would also make it harder to get the couple million people to start with.

Edit: According to Claude you're actually talking more like several hundred million patients, i.e. more Alzheimer's patients than exist, to detect a single ingredient's effect.

Considering that regular supermarket foodstuffs contain hundreds of thousands of substances, most having no effect, but some harmful and others helpful, and people eat different diets fairly randomly, we are already doing this experiment, we just aren't collecting the results.