Right, found this myself after a quick Google search, correct me if I'm wrong. It's not the density or temperature of the plasma that really affect transparency, rather, it's the fact that this plasma is opaque only because there exist photons of high enough energy to rip apart nuclei and electrons again (the electrons can then scatter incoming photons galore). As the expansion of the universe redshifts the photons, at a certain point there will be no photons of sufficient energy to ionise atoms, hence simultaneity.

Follow up question: assuming we're talking about a Boltzmann distribution, at what timescales do we expect to go from e.g. 1% to 99% of photons falling below ionisation energy?

How simultaneously did the universe go from opaque to transparent? I'd imagine that the different CMB temperatures correspond to different densities and therefore different temperatures, so is "at the same time" just more or less the same time on a cosmic scale?