The rights would belong to the person or company who purchased the rights and they would then be owed the royalties.
Doesn’t work like that. The singer is paid by one of a couple of companies for the public use rights. That is to play in elevators, as bumper music, in the restaurant or barber shop. If the politician, barber, or restauranteur has paid the fee to that company, everything in their songbook is fair game. The artist can no longer pick and choose and neither can the service if it was in their catalogue. They have no case.
But thanks for playing.