The X-ray-Weak Tail of the Quasar Population: Physical Properties and Redshift Evolution of Variability-Selected AGN in eFEDS

(abridged) Aims: To characterize variability-selected AGN candidates within the eROSITA Final Equatorial-Depth Survey (eFEDS), leveraging the synergy between optical variability and deep X-ray observations by SRG (Spectrum Roentgen Gamma)/eROSITA, to compare the physical properties of AGN populations selected through complementary methods. Methods: We used a sample of AGN candidates selected via optical variability by Ar\'evalo et al. and cross-match it to the X-ray selected eFEDS AGN sample. Spectroscopic characterizations are obtained for the matched samples from the Dark Energy Spectroscopic Instrument public data products. Results: We find a high classification purity with 98% candidates confirmed as broad-line AGN/QSOs. We identify a significant population (25%) that lack X-ray counterparts (X-U) in the eFEDS survey. We find that the X-D and X-U samples exhibit distinct physical properties and redshift distributions. The X-U sample peaks at $z = 2$ while the X-D sample at $z =1$, with nearly half of the variability-selected AGN at $z > 2.5$ remaining undetected in X-rays. This missing fraction can be attributed to K-correction effects and SED evolution with luminosity, mainly resulting in high accretion rate objects falling out of the X-D sample at higher redshifts. X-ray variability accounts for a part of the missing fraction but its dependence with redshift is less clear. Ensemble spectral analysis reveals that X-D sources have steeper optical continua and stronger broad emission lines than their X-U counterparts, while the forbidden and semi-forbidden lines have equal EWs in both samples. Finally, broad absorption line (BAL) features are found almost exclusively in the X-U population. This "missing" X-ray population must be accounted for when extrapolating X-ray-selected AGN to the broader Type I population and could impact cosmic estimates of massive black hole growth.