To study the demographics of interstellar ices in the Central Molecular Zone (CMZ) of the Milky Way, we obtain near-infrared spectra of 109 red point sources using NASA IRTF/SpeX at Maunakea. We select the sample from near- and mid-infrared photometry, including 12 objects in the previous paper of this series, to ensure that these sources trace a large amount of absorption through clouds in each line of sight. We find that most of the sample (100 objects) show CO band-head absorption at 2.3{mu}m, tagging them as red (super-) giants. Despite the photospheric signature, however, a fraction of the sample with L-band spectra (9/82=0.11) exhibit large H_2_O ice column densities (N>2x10^18^cm^-2^), and six of them also reveal CH_3_OH ice absorption. As one of such objects is identified as a young stellar object (YSO) in our previous work, these ice-rich sight lines are likely associated with background stars in projection to an extended envelope of a YSO or a dense cloud core. The low frequency of such objects in the early stage of stellar evolution implies a low star-formation rate (<~0.02M_{sun}_/yr), reinforcing the previous claim on the suppressed star-formation activity in the CMZ. Our data also indicate that the strong "shoulder" CO_2_ ice absorption at 15.4{mu}m observed in YSO candidates in the previous paper arises from CH3OH-rich ice grains having a large CO2 concentration [N(CO_2_)/N(CH_3_OH)~1/3].