That has actually been done in towns where the police chief couldn't pass the qualifications. Haha. Happenend in Portland, OR, IIRC many years ago.
The sheriff in our county sees herself as more of an administrator. She couldn't pass the certification requirements most police officers have to meet. Since she is sheriff, she is technically exempt by statute but she doesn't carry (for liability reasons I believe). She has actually been a pretty good sheriff and will be reelected in November.