There seems to be some confussion about "Chain Guns"....
The chain gun is an externally powered gun that uses a chain, much like a bicycle chain, to move the bolt back and forth to cycle the action. They only have one barrel, and, actually, have very moderate rates of fire, 350 to 600 rpm.
They were developed by Hughes back in the early 1970s.
There are many types of 'externally powered' machine gun, some have a single barrel, some have multiple barrels. The reason you have multiple barrels is the simple fact that the time required to move the bolt back far enough the extract and eject the spent cartridge, feed, lock and fire a new cartridge limits the rate of fire to around 1500 rpm. If you want a faster rate of fire, you need to be doing some of these operations at the same time, which means you need multiple barrels (Gatling type), or multiple chambers (revolver type). Also, at 1500 rpm, the stresses on the the case, and reciprocating parts are extremely high, In order to make a lighter gun, you need to slow the speed down, but to keep the high RoF, loading and firing need to be done simultaneously.
Also, the M61A1, GAU-8, GAU-12, M197, etc are externally powered, but all Gatling type machine guns are not necessarily externally powered, the Russian AK-630 is a self-powered Gatling type machine gun with a RoF in the 6000 rpm range, as was the gun in the SUU-23/A gun pod (a modified M61A1).
The GSh-23 is another example of a multiple barrel, self-powered (gas) machine gun, this time only two barrels, with a very high RoF, around 3500 rpm.