[I.e. if you assumed quantum mechanics didn't exist and that Maxwell's equations were the ground truth, you could explain this behaviour without any issue (with some leeway to define a polarizer)].
I think if you try and use that line of thought strictly (only call an explanation "classical" if you can explain with discrete photon particles), you'd probably have to argue that basically all of electromagnetism is fundamentally quantum mechanical. Again, while not strictly wrong (given our present knowledge), this goes too far for me (and I'd imagine most people).
Also: The "three-polarizer" experiment from your quoted video has a perfectly simple explanation in terms of electromagnetic waves. You use a polarizer to get light polarized along say "y" == (0,1). If you put another polarizer in front of it along "x" == (1,0) then the projection is of the E-field is zero and no light passes through. Now add another polarizer at 45 deg between the two: it then projects the E-field onto its axis, mapping it from (0,E) to (E/2,E/2) (magnitude is 1/sqrt(2)). The E-field now has a non-zero component along "x". So light comes out the final polarizer.
1. Take two of the lenses, hold them an inch apart, and shine a light so that it goes through both. The amount of light that goes through depends on their relative angle; at the right angle (90 degrees difference), no light will pass through. Hold them like this, so that no light gets through.
2. Insert a third polarizing filter between the two at a 45 degree angle. Amazingly, some light will now get through. You added an obstacle, and more light passes through.
I want to ask, though. Is it correct to call the third filter an obstacle? In the quantum realm, it’s not really an obstacle, is it?
Evidently not!
Where goes the energy of the orthogonal component of the field? Absorbed by the polarizer, reflected, ... ?
For example, a simple polarizer could be a grid of thin metal wires whose spacing is smaller than the wave-length of the incoming light. For the component of the E-field parallel to the wires currents can be induced freely along their length, and so the grid behaves much like a solid metal plate and reflects that part of the wave. For the component of the E-field perpendicular to the wires, significant currents can't be generated (since the wires are thin) and that part of the wave passes through.
[1] https://en.wikipedia.org/wiki/Polarizer
[2] https://en.wikipedia.org/wiki/Polarizer#Wire-grid_polarizers
It depends on the type of polariser.
The type used in LCD displays and 3D cinema glasses absorbs, that's why everything looks darker through them but they don't look like mirrors.
A polarising beam splitter reflects one mode and passes the other. It looks like a half-mirror.
Polarizers essentially just project the electric field of the wave onto some axis, zeroing out the perpendicular component. Keeping in mind that light intensity is the square of the electric field strength, all of this can be explained through straightforward classical electrodynamics.
An analogous statement would be that interference of light (say through a pair of slits) is also quantum mechanical in nature. This isn't strictly wrong (since basically everything is quantum mechanical in nature when you get down to it) but is a misleading way to present something that can (and was) understood perfectly well before quantum mechanics came along.
Note: These kind of experiments for a single particle (e.g. photons, electrons, etc) are a different story and do provide a demonstration of quantum mechanics (and the combination of wave-like and particle-like properties intrinsic to it).