ROM Manager and TWRP Problems on new CyanogenMod 13 Installation

After installing CyanogenMod 13 on my Nexus 5, ROM Manager failed to back up the current ROM or reboot into recovery. Both of these actions just rebooted the phone. It was possible to manually reboot into recovery and perform a backup, but ROM Manager could not see it. The problem was that although root access was enabled in Privacy GuardĀ for ROM Manager, it was not enabled for the OS in Settings -> Developer Options.