My guess: Zinc isn’t patentable, nobody can make money from selling it so nobody will do research to see if it works.
While that might be what’s driving it there are ways around it. Zinc isn’t directly patentable but you could patent a specific way to package it.
I could imagine packing it in Lozenges with copper might be a good idea. Taking a lot of Zinc reduces your copper absorption, so that you need to consume more copper when taking a lot of Zinc. It’s also possible that the copper helps directly.
But wouldn’t governments have a big incentive to do this? Even for private companies, it can bring them a lot of goodwill for presumably not that much money …
Trump appears to have a big incentive to do this.
Companies with experience at running clinical trials are probably busy testing more profitable treatments that would bring almost as much goodwill.