There's some good info here but also a lot of misleading or easily misread info.

You're correct on the fungibility of photons. Nothing to add there.

But you are dangerously vague for the rest.

> The easiest to understand model is the one you mentioned- and it does work.

Careful. The confusion you describe (even among physicists) comes at the root from not being specific on what we mean by absorption and emission, and unless you clarify you have not helped the situation. Let me try to be extremely clear about the possible models.

1. Classical scattering of waves. Here, there is a main EM wave which can partially scatter off atoms, since they are charged -- and importantly it's a collective scattering from all of the atoms in the path. The superposition (combination) of the original wave and the small scattered waves leads to a new, slower wave, with the same frequency as the original. Note that at no point is the incoming wave completely extinguished (absorbed), it's only ever a partial effect. This model works mathematically.

2. Classical scattering of particles. Here, you imagine a billiard ball photon travelling along that happens to hit an atom and be absorbed. For a moment there is no more photon anywhere. Then the atom relaxes, and a photon is re-emitted to continue on its journey. This model does not work mathematically in any way to explain the speed of light in a medium. You cannot assign the slowing down of light to a random time delay between classical particle absorption and emission nor to a particle taking a 'longer zigzag route' through the medium.
   When people ask this question on the internet, it's usually with a foundation of only classical mechanics, not quantum mechanics, and so this is usually what they are imagining when they say "absorption and re-emission" and it is wrong.

3. Quantum mechanical scattering. This one is tricky to understand without at least an intro to quantum mechanics but fundamentally we are scattering probabilities, not whole particles. There are several ways to do the maths -- you can consider the propagation, partial scattering, and interference of the photon wavefunction as it interacts with virtual energy levels in the atoms (looks very similar to the classical wave math -- superpositions of the original and partially scattered probability waves). Virtual energy levels are guaranteed to be unstable and exciting the atom to one is fundamentally different to exciting to a stable energy level. Or you can consider a path integral approach like that of Feynman (again, summing probable paths, not definite or discrete ones), or you can start calculating the collective excitation of the photon and all neighbouring atoms as a dressed quasiparticle with new properties (specifically, mass -- so travels less than c). The important thing here is that all the behaviour is collective, never discretely with a single atom -- and at no point is the photon (with its original frequency and direction) completely gone. When physicists say "absorption and re-emission" they are often thinking of this model because we use the same terms for classical scattering and scattering of a QM wavefunction -- but that does not mean it is the same model as 2) above. It is not, it is fundamentally different.

> The most common complaint is that an atom can only absorb very specific wavelengths, but light of all wavelengths is slowed down by materials. But, this is handled by understanding that collections of particles will have nearly an infinite number of modes of excitation

Your answer doesn't work. It is true that "excitations are always discrete" is an oversimplification -- the existence of black-body radiation proves that. But you don't get to claim that all materials absorb at all frequencies and therefore slow light through absorption and reemission, without also explaining why the transparency of a material doesn't have anything to do with the speed of light through it (cf. glass at optical wavelengths). Again this confusion you introduced comes from not clearly differentiating between the QM scattering of probabilities -- which may involve virtual energy levels and whole lot of behind-the-scenes stuff -- and complete, classical absorption of a photon to a new stable energy level. The real answer is simply that you don't need stable energy levels (which cause the material to become opaque) to exist to do the QM scattering math. Though transitions to virtual energy levels are low probability and necessarily temporary, their collective sum, including the original photon as well, gets us to where we want to go.