Spectroscopy of X-ray emission lines emitted in accretion discs around supermassive black holes is one of the most powerful probes of the accretion flow physics and geometry, while also providing in principle observational constraints on the black hole spin.[...] We aim at determining the ultimate physical driver of the strength of this relativistic reprocessing feature. We first extend the hard X-ray flux-limited sample of Seyfert galaxies studied so far (FERO, de la Calle Perez et al. 2010) to obscured objects up to a column density N_H=6x10^23 atoms/cm/cm. We verify that none of the line properties depends on the AGN optical classification, as expected from the Seyfert unification scenarios. There is also no correlation between the accretion disc inclination, as derived from formal fits of the line profiles, and the optical type or host galaxy aspect angle, suggesting that the innermost regions of the accretion disc and the host galaxy plane are not aligned. [...]. Data are not sensitive enough to the detailed ionisation state of the line-emitting disc. However, the lack of dependency of the line EW on either the luminosity or the rest-frame centroid energy rules out that disc ionisation plays an important role on the EW dynamical range in Seyferts. The dynamical range of the relativistically broadened K-alpha iron line EW in nearby Seyferts appears to be mainly determined by the properties of the innermost accretion flow. We discuss several mechanisms (disc ionisation, disc truncation, aberration due to a mildly relativistic outflowing corona) which can explain this. [...] Observational data are still not in contradiction with scenarios invoking different mechanisms for the spectral complexity around the iron line, most notably the partial covering absorption scenario. (abridged).
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