Engines, thank you for the recap of QWI course air-to-air gunnery phase day one. I concur. And that is my point. I lost my last attempted response thanks to an unfortunate Mr Apple Pk failure. So forgive me if I'm brief this time.
My original point was about round calibre and rate of fire. You now move on to the types of gun and, I hope most here will appreciate the differences, advantages and disadvantages of auto/revolver/Gatling. We haven't touched modern sighting systems/AI radar radars and the accuracy and relative ease of use.
Muzzle velocity is clearly a factor, but eventually that, in VERY SIMPLE TERMS becomes a sighting and range issue.
You have burst vs round correct. Ph and Pk for an individual round is a variable feast. Hit what, for example? The airframe, a vital system, a void? The comparison I made was between more, smaller calibre HE rounds and fewer, larger calibre HE rounds. The end result of which is potential for a spread of positive Ph rounds with a probability of damaging more vital organs (for example multiple routings of FBW lines) against fewer, larger rounds with a larger damage radius, but with the risk of leaving some redundancy functioning.
I like your maths, but practical experience doesn't always follow the models. My experience with air-to-air practice gunnery, live gunnery trials and test firing against test targets leads me to conclude the difference in requirements for AA and AG. That's all I'm saying.