Astro2020 Science White Paper: The Extended Cool Gas Reservoirs Within z > 1 (Proto-)Cluster Environments

High-redshift ($z$) proto-clusters will serve as testing grounds to probe the gas supply furnishing the emerging metals, stars, and large-scale structures we see at the current epoch. This work focuses on the major role large radio/millimeter (mm) single dish facilities will have in constraining the bulk, cold (T $= 10^{1-4}$K) molecular and atomic gas content. To highlight the need for large radio/mm single dishes, we calculate how the high-sensitivity of the Green Bank Telescope's (GBT) unblocked 100m aperture provides vital interferometric short-spacing coverage to support higher-resolution ngVLA observations of the cold neutral gas at the largest scales. These combined observations are optimal for revealing low-surface brightness emission, and thus aid in the total baryonic mass estimates across cosmic time.