[Edit]In the second video above, he notes that phase kicks don't change the speed of the wave, even though the wavelengths can stretch out, it's still lower than the speed of light in vacuum.
If you want an actual non-layman explanation for why light slows down ib a medium you need to express the light and medium interaction in terms of polaritons. Of course this is incredibly difficult to do so for even simple cases, so alas a whole suite of simplified explanations exist that seek to explain some aspect of the situation while failing to explain others.
"Phase kicks" explain the phase velocity. The speed of a wave is the group velocity. Phase velocity is omega(k)/k, group velocity is omega'(k). So if "phase kicks" explain phase velocity sufficiently so it's valid for a range of frequencies, then it also explains group velocity by taking the derivative of the dispersion relation.
The group velocity of these pulses through a medium will still be lower than the speed of light in a vacuum and phase kicks won't influence the group velocity.
Explaining how the group velocity of light can be slower in a medium than in a vacuum requires analyzing the coupling of photons with electrons to form polariton quasiparticles. You can then calculate the mass of these polaritons which in turn gives you the speed and get the full picture. Doing this, however, is incredibly complex and so it's much easier to consider simplified scenarios like either the steady-state case where you can simply reason about the scenario in terms of interference patterns between the light wave and the electromagnetic waves produced by oscillating electrons, or you can consider some non-steady state scenarios involving photons themselves being absorbed and reemitted by electrons but neither of these explanations fully capture the phenomenon.
I agree that the phase-kick classical model is a very simplified material model, which probably breaks down in certain ways. But it does yield a dispersion relation, therefore both phase and group velocities.
I can imagine things get really weird with pulses, but that's out of scope for the video.
So, there no slowdown by phase shift at all? But.. are you sure, because it looks like the phase shift is the real effect and logically speaking result of the phase shift will be the effect that look like slowdown for outside observer.
Also, how can you simulate similar effect in water and sound if there no sound polarization?
But this model won't be useful for understanding pulses of light through a medium. For example if I emit a very short pulse of light through a medium and measure how long it takes the head of the pulse to traverse the medium, using this model won't work. A phase shift won't move the head of a pulse of light backwards, it will just adjust the amplitude of the head as it propagates through the medium.
You still need a way to explain how it is that the head of the pulse actually takes longer to travel through the medium and that's not something a phase shift can explain.
If there was a phase shift but no slowdown, wouldn't we expect individual photons to propagate through a 10km loop of fibre optic cable without any slowdown, which would be measurable?