The original list of salicylates used by Dr. Feingold was designed for people with aspirin-induced asthma. For those whose asthma is made worse by aspirin, avoiding the entire list of salicylate-containing foods is beneficial.
Moreover, it has long been known that at least some of the food dyes appear to "cross-sensitize" with aspirin even though they are not chemically similar. Many of the colorings, moreover, are bronchoconstrictors, narrowing the bronchial tubes and preparing them to be more sensitive to whatever the next allergen is (e.g., pollen) that comes along. Even people with asthma not thought to be aspirin-induced have reported benefits from the Feingold diet.
When Dr. Feingold first began to eliminate the artificial flavorings, his reasoning was that if food dyes can cross-sensitize with aspirin, surely some of these unknown and untested flavorings can, too. There are thousands of flavorings in the American food supply - many times more than there were in Dr. Feingold's day - and yet almost none of them are tested for safety, let alone for neurotoxicity, bronchoconstricting properties, etc. While some of these vast numbers of chemicals are probably completely safe, it is almost impossible to identify them, or even to know which ones are in which foods. Most are simply listed as "artificial flavors."
Links: - Studies on PST, Salicylates, and Sulfation