It seemed too weird but I downloaded a trial of Logic Express and sure enough, I experienced the exact same behavior. After a bit of poking around I discovered what was happening. Logic Pro/Express stops all launchd jobs. Hazel uses launchd to start its background processes so it was a bit disconcerting to see another program, especially one from Apple, disabling yours on purpose, albeit indirectly. At least Logic is nice enough to start the jobs again when it quits.
That doesn’t seem right.