The 21-cm power spectrum of reionization is a promising probe for cosmology and fundamental physics. Exploiting this new observable, however, requires fast predictors capable of efficiently scanning the very large parameter space of cosmological and astrophysical uncertainties. In this paper, we introduce the halo model of reionization (HMreio), a new analytical tool that combines the halo model of the cosmic dawn with the excursion-set bubble model for reionization, assuming an empirical correction factor to deal with overlapping ionization bubbles. First, HMreio is validated against results from the well-known seminumerical code 21 cmfast, showing a good overall agreement for wave-modes of k≲1 h/Mpc. Based on this result, we perform a Monte-Carlo Markov-Chain (MCMC) forecast analysis assuming mock data from 1000-hour observations with the low-frequency part of the Square Kilometre Array (SKA) observatory. We simultaneously vary the six standard cosmological parameters together with seven astrophysical parameters quantifying the abundance and spectral properties of sources. Depending on the assumed theory error, we find very competitive constraints on cosmological parameters. In particular, it will be possible to conclusively test current cosmological tensions related to the Hubble parameter (H0-tension) and the matter clustering amplitude (S8-tension). Furthermore, the sum of the neutrino masses can be strongly constrained, making it possible to determine the neutrino mass hierarchy at the ∼90 percent confidence level. However, these goals can only be achieved if the current modeling uncertainties are substantially reduced to below ∼3 percent.