  The measurements of the 21-cm brightness temperature fluctuations from the
neutral hydrogen at the Epoch of Reionization (EoR) should inaugurate the next
generation of cosmological observables. In this respect, many works have
concentrated on the disambiguation of the cosmological signals from the
dominant reionization foregrounds. However, even after perfect foregrounds
removal, our ignorance on the background reionization history can significantly
affect the cosmological parameter estimation. In particular, the
interdependence between the hydrogen ionized fraction, the baryon density and
the optical depth to the redshift of observation induce nontrivial degeneracies
between the cosmological parameters that have not been considered so far. Using
a simple, but consistent reionization model, we revisit their expected
constraints for a futuristic giant 21-cm omniscope by using for the first time
Markov Chain Monte Carlo (MCMC) methods on multiredshift full sky simulated
data. Our results agree well with the usual Fisher matrix analysis on the
three-dimensional flat sky power spectrum but only when the above-mentioned
degeneracies are kept under control. In the opposite situation, Fisher results
can be inaccurate. We show that these conditions can be fulfilled by combining
cosmic microwave background measurements with multiple observation redshifts
probing the beginning of EoR. This allows a precise reconstruction of the total
optical depth, reionization duration and maximal spin temperature. Finally, we
discuss the robustness of these results in presence of unresolved ionizing
sources. Although most of the standard cosmological parameters remain weakly
affected, we find a significant degradation of the background reionization
parameter estimation in presence of nuisance ionizing sources.
