Brown dwarfs are common as free-floating objects but rare as close companions to Sun-like stars, a disparity known as the "brown-dwarf desert". Host-star metallicity can constrain whether these companions form mainly through metal-sensitive core accretion or through less metal-dependent disc or cloud fragmentation. We extend our homogeneous spectroscopic analysis of directly imaged planet hosts into the brown-dwarf regime and compare their metallicities with those of planet hosts and close-in brown-dwarf hosts. We compiled 54 unique directly imaged brown-dwarf systems selected over an inclusive 13-80 M_Jup interval and projected separations from about 5 au to several thousand au. Objects near the model-dependent 70-75 M_Jup hydrogen-burning boundary may instead be very-low-mass stars. For 31 hosts with archival high-resolution spectra, we derived atmospheric parameters and metallicities using Bayesian spectral synthesis. Literature companion masses and projected separations are heterogeneous and are used only for demographic context. Galactic velocities were calculated for 46 hosts solely to characterise the youth-biased imaging sample. The host stars have a broadly solar metallicity distribution, with a median [Fe/H] of +0.06 dex and a median absolute deviation of 0.11 dex, and show no strong metal-rich bias. No statistically significant metallicity difference is detected between the lower- and higher-mass directly imaged subsamples. The hosts are kinematically cold, as expected from the youth-biased selection of direct-imaging surveys. The absence of a strong metal-rich bias suggests that classical core accretion does not dominate the wide-orbit brown-dwarf population. Disc instability and cloud fragmentation remain plausible, but the current sample and heterogeneous companion properties do not permit object-by-object discrimination between these channels.