N-body dark matter simulations of structure formation in the Λ cold dark matter (ΛCDM) model predict a population of subhaloes within Galactic haloes that have higher central densities than inferred for the Milky Way satellites, a tension known as the ‘too big to fail’ problem. Proposed solutions include baryonic effects, a smaller mass for the Milky Way halo and warm dark matter (WDM). We test these possibilities using a semi-analytic model of galaxy formation to generate luminosity functions for Milky Way halo-analogue satellite populations, the results of which are then coupled to the Jiang & van den Bosch model of subhalo stripping to predict the subhalo Vmax functions for the 10 brightest satellites. We find that selecting the brightest satellites (as opposed to the most massive) and modelling the expulsion of gas by supernovae at early times increases the likelihood of generating the observed Milky Way satellite Vmax function. The preferred halo mass is 6 × 1011 M⊙, which has a 14 per cent probability to host a Vmax function like that of the Milky Way satellites. We conclude that the Milky Way satellite Vmax function is compatible with a CDM cosmology, as previously found by Sawala et al. using hydrodynamic simulations. Sterile neutrino-WDM models achieve a higher degree of agreement with the observations, with a maximum 50 per cent chance of generating the observed Milky Way satellite Vmax function. However, more work is required to check that the semi-analytic stripping model is calibrated correctly for each sterile neutrino cosmology.