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Cross-sectional area is a function of diameter so it is kinda hard to understand how you can dismiss it and then say that a cross-sectional area is more to the point.
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It's not a question of "dismissing" it, rather a point that it's not the important factor in play. You can derive cross-sectional area starting with the radius as well ... it's just an innumerate way to look at the problem.
It also plays against a general poor understanding of physics and body structures. You aren't any more likely to be significantly harmed if stabbed with a 10ga needle than a 20ga needle, for instance ... even though one is well over
twice the diameter of the other. They are both within a range where the body's mechanisms deal with those wounds in about the same way.
People get hung up on trivial measurements which don't adequately describe the whole problem. What's a "better" projectile: .40S&W or 10mm? ... many will not realize that they are the
same danged bullet, loaded into different cartridges. Did the
diameter of the bullet tell us anything meaningful?
It's the same thing with the most common handgun rounds. They all produce wounds within such a small range of difference that they are essentially "the same thing", as far as the body is concerned. 9/40/45 are all attempts to balance and optimize various factors in a hand-sized weapon. If you go too far from those design boundaries in one direction or another to make some important "gain", you begin to lose something just as important somewhere else. That's one reason why all of the common calibers are within 2mm or so, diameter-wise.