Joking aside, I have them on The List. I have picked up various lines a few times and have been lucky to have sensed it in time and shake them off by judicious use of reverse (remnants of one incident shown in pic). Did have to have a diver remove a big wad of polypropelene we picked up somewhere coming into Daytona, we were lucky not to have had a bigger problem.
I know divers don't like them. For instance, Jeff Seigal of Active Captain fame once told me that he used to provide diving services in Maine, and that he had had to disentangle many a lobster pot line from a cutter-equipped boat. Other divers have said they make cleaning the props and shafts more difficult and dangerous. Perhaps also lurking in there is that cutters may be bad for business but most divers I know aren't that way.
Still, I'd feel a little better with than without, although I have (barely) gotten away without them for almost six years of cruising in waters rife with pot and net lines. My autopilot has the uncanny ability to raise previously-unseen floats directly in front of us every time I set a new course.