And many if not most of the artists who are complaining about the licensed use of their songs, gave up the right to control those uses when they signed various contracts with publishers, recording companies and licensing agents in return for the anticipated income they would receive. In cases where they have not sold the rights, public performance, and synchronization are not like recording a cover, there is no compulsory license available, and they could refuse to license individual uses.
US law has a pitiful implementation of moral rights that is limited in federal statute (17 U.S.C. §106A) to certain works of the visual arts, which can generally not be defaced by their owners. There is some protection under state laws and concepts like injury due to tarnishment of brand that might be used to try to enjoin this, but most of the time, these artists just want to keep the money and assert their objection publically rather than spend their money on a lawsuit of doubtful outcome.