I was under the impression that AK-630 family of CIWS were fully automated/automatic. Such that target tracking and firing are done via software. But from the video, it appears as though the shooter is actually able to manually manipulate the gun itself.
They are, but they have backup modes for targets the radars are not great at detecting and tracking... like small targets in the water like surfaced mines or divers or small boats.
These are not CIWS variety
AK-630 here is is manual with slower rate of fire and is used instead of heavy machine guns
They got the same setup on original three Buyan ships
There are lots of versions of the AK-630 and all are CIWS weapons.
The system I think you mean is the AK-306 which looks the same but has a rate of fire of 800-1,000 rpm and is not water cooled and is intended for lighter boats and is generally only controlled manually by the pedestal mount and the internal gunners position.
The AK-630 is generally linked to a computer system but can be used in a variety of ways including one gun mount controlled externally by a person on the pedestal aiming system for shooting things close to the ship, often in the water like a floating mine or diver or small boat. It can also be manually aimed using the internal position shown in the video above with video view of the target through night vision optics controlling both guns at one time... this would also be used to fire on targets on a beach when landing that are not easily detected by the radar system.
To defend the vessel against anti ship weapons the radars can direct the guns and fire on small fast moving targets in a CIWS role too.
The Duet also has two AK-630 guns, as do the Kashtan and Palma and Pantsir naval systems which fire about 5,000 rpm... the gun is related to the 30mm 6 barrel gatling gun fitted to the MiG-27 and to some aircraft carried gun pods.
The AK-306 is lighter and simpler and cheaper and when shooting at targets on the shore or in the water does not waste so much ammo with its lower rate of fire... but it is not so good against enemy aircraft.
The AK-630 has four times more ammo on the gun mount and fires at 5 times the rate of fire... 2000 rounds and 5,000 rpm (compared with 500 rounds and 1,000 rpm.