But guilty until proven innocent is what being suggested here, so you would need to check thousands of different compounds in say coffee and run trials on them all individually?

Unless you are making your coffee out of an chemical soup instead of coffee beans it would be sufficient to run trials on the beans.

And to pile on, many of those thousands of compounds are already known to be toxic and/or carcinogenic at high levels.

To be fair, that specific warning is (probably) due to a different carcinogenic food additive: heat

The burden of proof would be in the company that wants to sell it. So “you” wouldn’t have to (nor your tax dollars). The company who’s investing resources to bring the product to market would have to. (And it should go without saying that they should also have to fund an impartial third party in order to do that)