They still got more kills with the AIM-7s and AIM-9s however.
Sure they were unreliable, but they gave more reliable kills than guns.
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There is a way to pull it off, most rounds, even artillery, have very little filler for round mass.
This is because you need the round to "protect" the filler while in the chamber from the high pressures.
The easiest way to deal with this, is using a lower pressure system (as with the 20x82 rounds they had much lower pressure than the 20x139 used at the time) so they can use heavier rounds, at the cost of velocity. Another example is comparing the SBD GBU-39 to the Mk82. The GBU-39 uses basically fiberglass for its outer shell while the Mk82 is still an iron bomb. The GBU-39, however, is much lighter at almost half the mass (129kg) compared to its Mk82 partner (227kg) but carry about the same amount of filler (GBU-93 has 93kg while the Mk82 has 87kg).
The best system to deal with a thinner casing is to lower chamber pressure.
How would you do this? With a high low pressure system.
Think of it as a reverse rocket system, all the high pressure is kept within the case and chamber, but the gases bleed over time out the top of the case through holes.
This way, you can propel more filler, for less overall round mass with similar velocities.
If you want to propel the same round mass with even more filler at the same velocity, it's about the same (give or take 10-15%) amount of propellant.
There's more advantages as well, you can make the gun itself lighter since the barrel doesn't have to deal with as much pressure as it gradually expands evenly along the bore rather than a sharp dropoff.
It's already used in grenade launchers, the Germans during WWII tried using it for antitank guns (the 8cm PAW 600 comes to mind, there were also plans for a 10cm called the 10H64 but that never came to fruition) and post war in some anti sub launchers, but that's about it.
The same can be said of VTOL turbine aircraft. The USMC used Harriers in Iraq out of a football stadium and kept them well protected, it's also the way they use their Cobras, unlike the US Army who keeps them at airstrips, they have an Arming point close to the front lines with just a few trucks and personnel to load them up and send them off. Same can be said of the V-22.
But you have one thing, Aircraft Carriers (and LHDs in the case of the Harrier, V-22 and F-35B) basically do the same thing you're talking about but would need the ship close by to pull it off.
There's also the issue with jet aircraft of sucking in sea water at takeoff in the wake it would create and all of that that I'll get into if you want to discuss it further.