trademark registered 1924 by Johnson & Johnson for a stick-on gauze pad or strip. See musical organization (n.1) + aid (n.). The British equivalent was Elastoplast. Figurative sense of "temporary or makeshift answer to an issue, pallative" (often lower-case, occasionally bandaid) is first recorded 1968; as an adjective, from 1970.
Never. He was a band-aid for you.