Context. Spectroscopy of X-ray emission lines emitted from 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. Previous studies have suggested that relativistically broadened line profiles are fairly common in nearby unobscured Seyfert galaxies. Their strength, as parametrised by the equivalent width (EW) against the total underlying continuum, spans a range of almost two orders of magnitude. Aims. We attempt to determine the ultimate physical driver of the strength of this relativistic reprocessing feature. Methods. We first extend the hard X-ray flux-limited sample of Seyfert galaxies studied so far in the framework of the FERO (Finding Extreme Relativistic Objects) project to obscured objects up to a column density N H = 6 × 10 23 cm −2 . 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. We use this extended sample to study the dependence of the EW on various observables, and compare it with the predictions of Monte Carlo accretion-disc reprocessing simulation. Results. The behaviour of the EW as a function of disc inclination, shape of the intrinsic power-law nuclear continuum, or iron abundance does not agree with the simulation predictions. Our observational data are not sensitive enough to the detailed ionisation state of the line-emitting disc. However, the lack of dependence of the line EW on either the luminosity or the rest-frame centroid energy rules out disc ionisation playing an important role on the EW dynamical range in Seyferts.Conclusions. The dynamical range of the relativistically broadened K α 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, and aberration due to a mildly relativistic outflowing corona) that can explain this. We stress that the above results represent neither a falsification or proof of the relativistically blurring scenario. Observational data still do not contradict scenarios invoking different mechanisms for the spectral complexity around the iron line, most notably the "partial covering" absorption scenario.