Boundary value problem

How does the math account for what the plate is attached to?

That is exactly where theory meets reality, and it's covered heavily in Chapter 4, which is all about boundary conditions.

You can calculate your flexural rigidity perfectly all day long, but an engineer must define the boundary conditions to actually solve the equations.

Got it.

The textbook outlines three primary edge conditions.

Yes.

Kirchhoff's math was right all along.

You can legitimately combine the twisting and shearing into one effective shear force, which reduces the boundary conditions from three to two, and the equations solve perfectly.