  One of the surprising early findings with JWST has been the discovery of a
strong "roll-over" or a softening of the absorption edge of Ly$\alpha$ in a
large number of galaxies at ($z\gtrsim 6$), in addition to systematic offsets
from photometric redshift estimates and fundamental galaxy scaling relations.
This has been interpreted as damped Ly$\alpha$ absorption (DLA) wings from high
column densities of neutral atomic hydrogen (HI), signifying major gas
accretion events in the formation of these galaxies. To explore this new
phenomenon systematically, we assemble the JWST/NIRSpec PRImordial gas Mass
AssembLy (PRIMAL) legacy survey of 494 galaxies at $z=5.5-13.4$. We
characterize this benchmark sample in full and spectroscopically derive the
galaxy redshifts, metallicities, star-formation rates, and ultraviolet slopes.
We define a new diagnostic, the Ly$\alpha$ damping parameter $D_{\rm Ly\alpha}$
to measure and quantify the Ly$\alpha$ emission strength, HI fraction in the
IGM, or local HI column density for each source. The JWST-PRIMAL survey is
based on the spectroscopic DAWN JWST Archive (DJA-Spec). All the software,
reduced spectra, and spectroscopically derived quantities and catalogs are made
publicly available in dedicated repositories. The fraction of strong galaxy
DLAs are found to be in the range $65-95\%$ at $z>5.5$. The fraction of strong
Ly$\alpha$ emitters (LAEs) is found to increase with decreasing redshift, in
qualitative agreement with previous observational results, and are
predominantly associated with low-metallicity and UV faint galaxies. By
contrast, strong DLAs are observed in galaxies with a variety of intrinsic
physical properties. Our results indicate that strong DLAs likely reflect a
particular early assembly phase of reionization-era galaxies, at which point
they are largely dominated by pristine HI gas accretion. [abridged]
