The progenitors of SN 2008S and the 2008 transient in NGC300 were dust-enshrouded, with extremely red mid-infrared (MIR) colors and relatively low luminosities. The transients were optically faint (-13 < M_V < -15) compared to normal core-collapse supernovae (SNe), and their spectra exhibited narrow emission lines. These events are unique among transient-progenitor pairs and hence constitute a new class. Whether they are true SNe or bright massive-star eruptions, we argue that their rate is ~20% of the SN rate. This fact is remarkable in light of the observation that a very small fraction of all massive stars have the MIR colors of the SN 2008S and NGC300 progenitors, as we show using MIR and optical luminosity, color, and variability properties of massive stars in M33. We find that the fraction of massive stars with colors consistent with these progenitors is 1/10000. In fact, only < 10 similar objects exist in M33 - all of which lie at the luminous red extremum of the AGB sequence. That these transients are relatively common with respect to SNe, while their progenitors are rare compared to the massive star population, implies that the dust-enshrouded phase is a short-lived phase in the lives of many massive stars. This shrouded epoch can occur only in the last ~10^4 yr before explosion, be it death or merely eruption. We discuss the implications of this finding for the evolution and census of ``low-mass'' massive stars (8-12 Msun), and we connect it with theoretical discussions of electron-capture SNe and the explosive birth of white dwarfs. A systematic census with (warm) Spitzer of galaxies in the local universe for analogous progenitors would significantly improve our knowledge of this channel to massive stellar explosions, and potentially to others with obscured progenitors. (Abridged)