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Combining 66 ultraviolet (UV) spectra and ancillary data from the Low-Redshift Lyman Continuum Survey (LzLCS) and 23 LyC observations by earlier studies, we form a statistical sample of star-forming galaxies at $z \sim 0.3$ to study the role of the cold interstellar medium (ISM) gas in the leakage of ionizing radiation. We first constrain the massive star content (ages and metallicities) and UV attenuation, by fitting the stellar continuum with a combination of simple stellar population models. The models, together with accurate LyC flux measurements, allow to determine the absolute LyC photon escape fraction for each galaxy ($f_{\rm esc}^{\rm abs}$). We measure the equivalent widths and residual fluxes of multiple HI and low-ionization state (LIS) lines, and the geometrical covering fraction adopting the picket-fence model. The $f_{\rm esc}^{\rm abs}$ spans a wide range, with a median (0.16, 0.84 quantiles) of 0.04 (0.02, 0.20), and 50 out of the 89 galaxies detected in the LyC. The HI and LIS line equivalent widths scale with the UV luminosity and attenuation, and inversely with the residual flux of the lines. The HI and LIS residual fluxes are correlated, indicating that the neutral gas is spatially traced by the LIS transitions. We find the observed trends of the absorption lines and the UV attenuation are primarily driven by the covering fraction. The non-uniform gas coverage demonstrates that LyC photons escape through low-column density channels in the ISM. The equivalent widths and residual fluxes of the UV lines strongly correlate with $f_{\rm esc}^{\rm abs}$: strong LyC leakers show weak absorption lines, low UV attenuation, and large Ly$α$ equivalent widths. We finally show that simultaneous UV absorption line and dust attenuation measurements can predict, on average, the escape fraction of galaxies and the method can be applied to galaxies across a wide redshift range.