In this paper we study the molecular gas content of a representative sample of 67 of the most massive early-type galaxies in the local universe, drawn uniformly from the MASSIVE survey. We present new IRAM-30m telescope observations of 30 of these galaxies, allowing us to probe the molecular gas content of the entire sample to a fixed molecular-to-stellar mass fraction of 0.1%. The total detection rate in this representative sample is 25$^{+5.9}_{-4.4}$%, and by combining the MASSIVE and ATLAS$^{rm 3D}$ molecular gas surveys we find a joint detection rate of 22.4$^{+2.4}_{-2.1}$%. This detection rate seems to be independent of galaxy mass, size, position on the fundamental plane, and local environment. We show here for the first time that true slow rotators can host molecular gas reservoirs, but the rate at which they do so is significantly lower than for fast-rotators. Objects with a higher velocity dispersion at fixed mass (a higher kinematic bulge fraction) are less likely to have detectable molecular gas, and where gas does exist, have lower molecular gas fractions. In addition, satellite galaxies in dense environments have $approx$0.6 dex lower molecular gas-to-stellar mass ratios than isolated objects. In order to interpret these results we created a toy model, which we use to constrain the origin of the gas in these systems. We are able to derive an independent estimate of the gas-rich merger rate in the low-redshift universe. These gas rich mergers appear to dominate the supply of gas to ETGs, but stellar mass loss, hot halo cooling and transformation of spiral galaxies also play a secondary role.
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