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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 pointsources using NASA IRTF/SpeX at Maunakea. We select the sample from near- andmid-infrared photometry, including $12$ objects in the previous paper of thisseries, to ensure that these sources trace a large amount of absorption throughclouds 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 samplewith $L$-band spectra ($9/82=0.11$) exhibit large H$_2$O ice column densities($N > 2\times10^{18}\ {\rm cm}^{-2}$), and six of them also reveal CH$_3$OH iceabsorption. As one of such objects is identified as a young stellar object(YSO) in our previous work, these ice-rich sight lines are likely associatedwith background stars in projection to an extended envelope of a YSO or a densecloud core. The low frequency of such objects in the early stage of stellarevolution implies a low star-formation rate ($<0.02\ M_\odot$ yr$^{-1}$),reinforcing the previous claim on the suppressed star-formation activity in theCMZ. 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 fromCH$_3$OH-rich ice grains having a large CO$_2$ concentration [$N {\rm (CO_2)} /N {\rm (CH_3OH)} \approx 1/3$].

Massive Young Stellar Objects in the Galactic Center. II. Seeing Through the Ice-rich Envelopes