We present the host-galaxy molecular gas properties of a sample of 213 nearby (0.01 < z < 0.05) hard-X-rayselected active galactic nucleus (AGN) galaxies, drawn from the 70-month catalog of Swift's Burst Alert Telescope (BAT), with 200 new CO(2-1) line measurements obtained with the James Clerk Maxwell Telescope and the Atacama Pathfinder Experiment telescope. We find that AGN in massive galaxies (log(M∗ M⊙) > 10.5) tend to have more molecular gas and higher gas fractions than inactive galaxies matched in stellar mass. When matched in star formation, we find AGN galaxies show no difference from inactive galaxies, with no evidence that AGN feedback affects the molecular gas. The higher molecular gas content is related to AGN galaxies hosting a population of gas-rich early types with an order of magnitude more molecular gas and a smaller fraction of quenched, passive galaxies (~5% versus 49%) compared to inactive galaxies. The likelihood of a given galaxy hosting an AGN (Lbol> 1044 erg s-1 ) increases by ~10-100 between a molecular gas mass of 108.7M⊙ and 1010.2M⊙. AGN galaxies with a higher Eddington ratio (log(L/LEdd) > -1.3) tend to have higher molecular gas masses and gas fractions. The log(NH/ cm-2 ) > 23.4) of AGN galaxies with higher column densities are associated with lower depletion timescales and may prefer hosts with more gas centrally concentrated in the bulge that may be more prone to quenching than galaxy-wide molecular gas. The significant average link of host-galaxy molecular gas supply to supermassive black hole (SMBH) growth may naturally lead to the general correlations found between SMBHs and their host galaxies, such as the correlations between SMBH mass and bulge properties, and the redshift evolution of star formation and SMBH growth.