  A measurement of a primordial non-Gaussianity (PNG) signal through late- or
early-Universe probes has the potential to transform our understanding of the
physics of the primordial Universe. While large-scale structure observables in
principle contain vital information, interpreting these measurements is
challenging due to poorly understood astrophysical effects. Luckily, $N$-body
simulations, such as the public \textsc{AbacusPNG} set presented in this study,
consisting of 9 boxes, each of size $L_{\rm box} = 2~{\rm Gpc}/h$ and particle
mass of $1.01 \times 10^{10} \ M_\odot/h$, provide a viable path forward. As
validation, we find good agreement between the simulations and our expectations
from one-loop perturbation theory and the `separate universe' method for the
matter bispectrum, matter power spectrum and the halo bias parameter associated
with PNG, $b_\phi$. As a science application, we investigate the link between
halo assembly bias and $b_\phi$ for halo properties known to play a vital role
in accurately predicting galaxy clustering: concentration, shear (environment),
and accretion rate. We find a strong response for all three parameters,
suggesting that the connection between $b_\phi$ and the assembly history of
halos needs to be taken into account by future PNG analyses. We further perform
the first study of the $b_\phi$ parameter from fits to early DESI data of the
luminous red galaxy (LRG) and quasi-stellar object (QSO) samples and comment on
the effect on $f_{\rm NL}$ constraints for the allowed galaxy-halo models (note
that $\sigma [f_{\rm NL}] \propto \frac{\sigma [b_\phi]}{b_\phi}$). We find
that the error on $f_{\rm NL}$ is 21, 6, 22 for the LRGs at $z = 0.5$ and $z =
0.8$ and QSOs at $z = 1.4$, respectively, suggesting that a thorough
understanding of galaxy assembly bias is warranted so as to perform robust
high-precision analysis of local-type PNG with future surveys.
