So, in answer to the OP's question, external power driven guns certainly do solve some problems, but they introduce others. They're not a panacea, more of an alternative option. Throughout the Cold War, Soviet designers preferred self-driven guns, developing a range of gas-operated Gatling-style multibarrel cannons. This is notable because all Western Gatling type weapons (at least that I know of) are externally-powered. The little quad-barrel 12.7mm Gatling machine gun under the nose of the Hind-D is an example.
According to George M. Chinn's magisterial volume
The Machine Gun, autocannon development was neglected generally in the first half of the 20th century, due to a combination of lack of vision, international treaties, and low defense spending during the interwar period.
As for specific pros and cons of externally power-driven guns, it is definitely true that they don't get stopped by underpowered ammunition, and they can easily fire multiple types of ammunition that produce different amounts of power. This isn't really a huge advantage for an aircraft gun though, that's more something that an armored fighting vehicle can take advantage of. The M242 in a Bradley, for instance, can switch from armor piercing to high explosive rounds with the flick of a switch, and the fact that the armor-piercing round has a much lower recoil impulse and vastly different internal pressure curve is not a problem.
With that said, the 2A42 30mm autocannon in a BMP-2 is dual-feed and is self-powered and seems to work just fine, so clever engineers
can replicate these advantages well-enough in a self-powered system.
For aircraft, the big problem with an externally-powered weapon is that external power doesn't intrinsically make the gun shoot any faster. There is a chapter in Chinn's aforementioned book on the development of a new model of 20mm cannon by Melvin Johnson (who would later assist with the development of the early AR-10, which later evolved into the AR-15), who was trying to make a better autocannon than either the Oerlikon or Hispano-Suiza Brirkigt type weapons. One of the features he wanted was a higher rate of fire and he achieved that... technically. The problem he ran into is that as the rate of fire goes up in a linear-action cannon, the acceleration of the rounds by the feed system also goes up, and he was having problems with the ammunition disintegrating while it was fed.
Chinese/Russian 30x165mm ammunition sports a
massive set of double crimps where the case mouth is mashed into grooves into the projectile to keep them from separating prematurely for exactly this reason.
For this, and related reasons, the path forward after the war wasn't running more power into a conventional linear action, it was in designs like Gatlings and revolver cannons where the ammunition feeding, loading, and extraction steps can all be parallelized so that the peak stress on the moving parts and the ammunition is reduced. The limiting factor on rate of fire in WWII weapons was not the available power for running the breech mechanism of the gun, it was the strength of the moving parts and of the ammunition.
