> They move at the speed of light and thus time does not advance for them.
Isn't it more accurate to say that photons move at the speed of causality, when the medium is a pure vacuum? Because in some other medium like glass, the speed of light is slower than the speed of causality.
So my follow-up question is: do slower photons (such as those propagating through a fiber-optic strand, or water) then experience the advancement of time?
There is no such thing as slow photons, photons always travel at the speed of light.
When light enters a medium there are two mostly (but not entirely) equal ways to think about what happens, one is to view light as a purely electromagnetic wave that interacts with atoms and causes the atoms to oscillate. This oscillation produces its own electromagnetic wave that interferes with the original wave. The result of this interference will be an electromagnetic wave with the same frequency, same amplitude, and travelling in the same direction as the incoming light but shifted backwards and it's that shift backwards that gives the appearance of light slowing down.
That explanation is pretty good and accounts for almost everything except for the latency of light through a medium.
If that's what you want to model, then it's better to think of light as made up of photons instead of being a wave, and then when photons enter a material they no longer exist as independent particles but through a process of absorption and reemission by electrons in the material become particles called polaritons. Polaritons do have mass and hence travel slower than the speed of light.
Neither of these explanations are perfect, but the full explanation is ridiculously complicated and there's no suitable metaphor for it. If you are interested in knowing the edge latency of light through a medium, then the polariton explanation is appropriate. If you want to know the "bandwidth" explanation of light through a medium, then the wave explanation is appropriate.