I've been experimenting with it for a bit to figure out what does what. When I assign a ship to patrol a system it is manually controlled, not automated. It will say it's on a patrolling mission, fly to the system, then go idle and show no mission. I noticed it goes for repairs on its own, then returns to the system it was patrolling, so that's good. So now the only problem with a ship patrolling a system is that they go idle, which means they are still patrolling, but they show up on my idle ships tab. If I manually set a ship to patrol a mining base, it will actually say it's on a patrolling mission and circle the base. I guess it would be something the Dev team would have to fix. I just don't want to cycle though my idle ships and have to go though a bunch of ships that are patrolling a system
Patrolling an entire system is slightly different to patrolling a specific base or colony. A manually-controlled mission to patrol a system will not do much more than move to the system.
I would suggest instead assigning a patrol mission for a specific base or colony and also set the ships engagement stance (comma key) to "Engage System targets", which should be the default stance for patrol missions anyway. This means that the ship will continually patrol the assigned base or colony, but will respond to any threats in the same system.