Inspired by recent observations of the absence and presence of superconductivity in Bernal-stacked bilayer graphene (BBG) and BBG-WSe2 heterostructure, we investigate all possible competing instabilities driven by van Hove singularities (vHS) near the Fermi level in spin-degenerate and spin-split BBG. By performing a parquet renormalization group analysis in both systems, we identify an incommensurate intra-valley charge density wave (CDW) as the dominant instability in spin-degenerate BBG. Moreover, we find that lifting the spin degeneracy by the Ising spin-orbit coupling from the WSe2 monolayer or by applying an in-plane magnetic field restores superconductivity as the primary instability since intra-valley CDW is suppressed when the vHS are spin-polarized. We also discuss future experiments to confirm our proposed scenarios.