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.