  To understand how massive galaxies at high-$z$ co-evolve with enormous
reservoirs of halo gas, it is essential to study the coldest phase of the
circum-galactic medium (CGM), which directly relates to stellar growth. The
SUPERCOLD-CGM survey is the first statistical survey of cold molecular gas on
CGM scales. We present ALMA+ACA observations of CO(4-3) and continuum emission
from 10 Enormous Ly$\alpha$ Nebula (ELANe) around ultraluminous type-I QSOs at
$z\sim2$. We detect CO(4-3) in 100$\%$ of our targets, with 60$\%$ showing
extended CO on scales of 15$-$100 kpc. Q1228+3128 reveals the most extended
CO(4-3) reservoir of $\sim$100 kpc and is the only radio-loud target in our
sample. The CO reservoir is located along the radio axis, which could indicate
a link between the inner radio-jet and cold halo gas. For the other five
radio-quiet ELANe, four of them show extended CO(4-3) predominantly in the
direction of their companions. These extended CO(4-3) reservoirs identify
enrichment of the CGM, and may potentially contribute to widespread star
formation. However, there is no evidence from CO(4-3) for diffuse molecular gas
spread across the full extent of the Ly$\alpha$ nebulae. One target in our
sample (Q0107) shows significant evidence for a massive CO disk associated with
the QSO. Moreover, 70$\%$ of our QSO fields contain at least one CO companion,
two of which reveal extended CO emission outside the ELANe. Our results provide
insight into roles of both the cold CGM and companions in driving the early
evolution of massive galaxies.
