"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.