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When accreting X-ray pulsars (XRPs) undergo bright X-ray outbursts, their luminosity-dependent spectral and timing features can be analysed in detail. The XRP GRO J1750-27 recently underwent one of such episodes, during which it was observed with $NuSTAR$ and monitored with $NICER$. Such a data set is rarely available, as it samples the outburst over more than a month at a luminosity that is always exceeding ${\sim}5\times10^{37}\,$erg/s. This value is larger than the typical critical luminosity value, where a radiative shock is formed above the neutron star's surface. Our data analysis of the joint spectra returns a highly ($N_H\sim(5-8)\times10^{22}\,$cm$^{-2}$) absorbed spectrum showing a K$α$ iron line, a soft blackbody component likely originating from the inner edge of the accretion disk, and confirms the discovery of one of the deepest cyclotron lines, at a centroid energy of ${\sim}44\,$keV corresponding to a magnetic field strength of $4.7\times10^{12}\,$G. This value is independently supported by the best-fit physical model for spectral formation in accreting XRPs which, in agreement with recent findings, favours a distance of $14$ kpc and also reflects a bulk-Comptonization dominated accretion flow. Contrary to theoretical expectations and observational evidence from other similar sources, the pulse profiles as observed by $NICER$ through the outburst raise, peak and decay remain remarkably steady. The $NICER$ spectrum, including the iron K$α$ line best-fit parameters, also remain almost unchanged at all probed outburst stages, similar to the pulsed fraction behaviour. We argue that all these phenomena are linked and interpret them as resulting from a saturation effect of the accretion column's emission, which occurs in the high-luminosity regime.