I dont think that works. If i try the 5 ingredient set and get sick, i gained no information about ingrediences 6,7,8,9,10 (if there are 10 ingredients in total). If next i try set 3 and get sick, i have made 2 tries and still have 0 information. If ingredient 1 makes me sick i will need log(n) tries until i try the set that only contains ingredient 1. After that i have only one point of information because everything i tried so far was tainted by ingredient 1. I have to try everything again.

Anytime you get sick you gain no information. If you are very lucky and ingedient 1-x are all harmless, then you will eliminate multiple at once. But if 1 is harmless, 2 makes you sick, and the next 1000 ingredients are harmelss you will still have to try log(1002) times to eliminate ingredient 2 and then you know nothing about 3-1003.

A Binary search requires a ordered data set. Something like "if you react to X, you will also react to any X+1, X+2… X+n. Food is not ordered, you cant know if you react badly to bell peper because you reacted badly to whole grain wheat.

Why though? How does this force anybody to do anything? They will fine anybody 2500$ for handing this to a kid and call it a day. Maybe even move closer to the maximum fine because it is a deliberate breaking of the law.