  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
($N > 2\times10^{18}\ {\rm cm}^{-2}$), 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 ($<0.02\ M_\odot$ yr$^{-1}$),
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
CH$_3$OH-rich ice grains having a large CO$_2$ concentration [$N {\rm (CO_2)} /
N {\rm (CH_3OH)} \approx 1/3$].
