Bipolar transistors are more complex, but they still work by controlling the bulk geometry of moving charge.

None of this requires QM, except in the very simple sense that you need to know what a bandgap is and how doping changes it. Beyond that, the "mechanical" details of the charge dynamics are just calculus - even more crudely, very very small plumbing.

QM effects are only relevant when the simple space charge models start to break down at very small geometries. Your "mechanical" space charge model becomes soft, noisy, and more complicated. Essentially the plumbing develops waves and ripples that depend on the geometry, and the edges of the pools and pipes can start to leak. And that's where QM finally starts to make a real difference.