Recent neutron-diffraction experiments in honeycomb CrI₃ quasi-2D ferromagnets have evinced the existence of a gap at the Dirac point in their spin-wave spectra. The existence of this gap has been attributed to strong in-plane Dzyaloshinskii-Moriya or Kitaev (DM/K) interactions and suggested to set the stage for topologically protected edge states to sustain non-dissipative spin transport. We perform state-of-the-art simulations of the spin-wave spectra in monolayer CrI₃, based on time-dependent density-functional perturbation theory (TDDFpT) and fully accounting for spin-orbit couplings (SOC) from which DM/K interactions ultimately stem. While our results are in qualitative agreement with experiments, the computed TDDFpT magnon gap at the Dirac point is found to be 0.47 meV, roughly six times smaller than the most recent experimental estimates, so questioning that intralayer anisotropies alone can explain the observed gap. Lattice-dynamical calculations, performed within density-functional perturbation theory (DFpT), indicate that a substantial degeneracy and a strong coupling between vibrational and magnetic excitations exist in this system, providing a possible additional gap-opening mechanism in the spin-wave spectra. To pursue this path, we introduce an interacting magnon-phonon Hamiltonian featuring a linear coupling between lattice and spin fluctuations, enabled by the magnetic anisotropy induced by SOC. Upon determination of the relevant interaction constants by DFpT and supercell calculations, this model allows us to propose magnon-phonon interactions as an important microscopic mechanism responsible for the enhancement of the gap in the range of ≈4meV around the Dirac point of the CrI₃ monolayer.