This is one of those questions that you either shrug off as “one of those things” or it pesters and nags at you until you resolve it. After a few hours of research and reading, I think I figured it out.
Like many arms developments, this stems back to military use. During the Revolutionary War, the ‘Brown Bess’ musket was a common sight on the battlefield. In order to maximize impact during volley exchanges, Washington ordered his troops to load “four or eight” buckshot pellets behind their musketballs. In a .69 muzzleloader, the best use of space was four balls approximately .27″ in diameter.
Where it gets interesting is with the .72 muzzleloaders. These had larger bores so the .27″ shot was not efficiently packed in. But three .32″ spheres fit nicely inside the .72″ rifle.
When load development among the military became standardized (largely as a result of breechloaders become more common), the Army Quartermaster directed that all cartridges be loaded with ball and buckshot for maximum effect. Given the increased need for performance at range, it made sense to go with .32 caliber bullets in order to overcome the most air resistance while retaining sufficient velocity to cause fatal wounds.
I mocked this up using a CAD program called Fusion360 to confirm this hypothesis. It seems very likely that as the military turned to the Spencer pump shotgun and later the M1897 Trench, specifications for shotshells used the common 12-gauge configuration and the recipe used by soldiers for nearly a century at that point in large-bore firearms: 9 pellets of .32″, colloquially known by then as #00 buckshot.
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