ABSTRACT The quenching ‘maintenance’ and related ‘cooling flow’ problems are important in galaxies from Milky Way mass through clusters. We investigate this in haloes with masses ∼$10^{12}\!-\!10^{14}\, {\rm M}_{⊙ }$, using non-cosmological high-resolution hydrodynamic simulations with the FIRE-2 (Feedback In Realistic Environments) stellar feedback model. We specifically focus on physics present without AGN, and show that various proposed ‘non-AGN’ solution mechanisms in the literature, including Type Ia supernovae, shocked AGB winds, other forms of stellar feedback (e.g. cosmic rays), magnetic fields, Spitzer–Braginskii conduction, or ‘morphological quenching’ do not halt or substantially reduce cooling flows nor maintain ‘quenched’ galaxies in this mass range. We show that stellar feedback (including cosmic rays from SNe) alters the balance of cold/warm gas and the rate at which the cooled gas within the galaxy turns into stars, but not the net baryonic inflow. If anything, outflowing metals and dense gas promote additional cooling. Conduction is important only in the most massive haloes, as expected, but even at ∼$10^{14}\, {\rm M}_{⊙ }$ reduces inflow only by a factor ∼2 (owing to saturation effects and anisotropic suppression). Changing the morphology of the galaxies only slightly alters their Toomre-Q parameter, and has no effect on cooling (as expected), so has essentially no effect on cooling flows or maintaining quenching. This all supports the idea that additional physics, e.g. AGN feedback, must be important in massive galaxies.