Stars can be ripped apart by tidal forces in the vicinity of a massive black hole (MBH), causing luminous flares known as tidal disruption events (TDEs). These events could be contributing to the mass growth of intermediate-mass MBHs. New samples from transient surveys can provide useful information on this unexplored growth channel. This work aims to study the demographics of TDEs by modeling the coevolution of MBHs and their galactic environments in a cosmological framework. We use the semianalytic galaxy formation model L-Galaxies BH, which follows the evolution of galaxies as well as of MBHs, including multiple scenarios for MBH seeds and growth, spin evolution, and binary MBH dynamics. We associated time-dependent TDE rates with each MBH depending on the stellar environment, following the solutions to the 1D Fokker Planck equation solved with PHASEFLOW. Our model produces volumetric rates that are in agreement with the latest optical and previous X-ray samples. This agreement requires a high occupation fraction of nuclear star clusters with MBHs since these star reservoirs host the majority of TDEs at all mass regimes. We predict that TDE rates are an increasing function of MBH mass up to ~105.5 M⊙, beyond which the distribution flattens and eventually drops for > 107 M⊙. In general, volumetric rates are predicted to be redshift independent at z < 1. We discuss how the spin distribution of MBHs around the event horizon suppression can be constrained via TDE rates and the average contribution of TDEs to the MBH growth. In our work, the majority of low-mass galaxies host nuclear star clusters that have their loss-cone depleted by z = 0, explaining why TDEs are rare in these systems. This highlights how essential time-dependent TDE rates are for any model to be in good agreement with observations at all mass regimes.