I just drew a box around half of the teeth. But the rest of the gear cross-section, being an airplane part, was sculpted to put material only where it was needed. Ideally, the strain at any point should be the same as any other point.

So the MoI was not simply a stackup of some stock shapes.

It's not that the calculation was particularly difficult, it's just that calculus was needed, and one couldn't just copy the formula for a stock shape.

It was just clear to me that some engineers were formula-pluggers and others learned it at a deeper level.

Stock shapes were certainly handy to check the math with, however. Fit the gear into a stock shape envelope, and have the gear envelope a stock shape. Your calculation should fall in between the results of those shapes.

The purpose of calculating the MoI was to determine if the stops at the end of the screw were strong enough to stop the gearbox without shattering.