Spin bath polarization is the key to enhancing the sensitivity of quantum sensing and information processing. Significant effort has been invested in identifying the consequences of quantumness and its control for spin-bath polarization. Here, by contrast, we focus on the adverse role of quantum correlations (entanglement) in a spin bath that can impede its cooling in many realistic scenarios. We propose to remove this impediment by modified cooling schemes, incorporating probe-induced disentanglement via alternating, non-commuting probe-bath interactions, so as to suppress the buildup of quantum correlations in the bath. The resulting bath polarization is thereby exponentially enhanced. The underlying thermodynamic principles have far-reaching implications for quantum technological applications