  Lithium ortho-thiophosphate (Li$_3$PS$_4$) has emerged as a promising
candidate for solid-state-electrolyte batteries, thanks to its highly
conductive phases, cheap components, and large electrochemical stability range.
Nonetheless, the microscopic mechanisms of Li-ion transport in Li$_3$PS$_4$ are
far to be fully understood, the role of PS$_4$ dynamics in charge transport
still being controversial. In this work, we build machine learning potentials
targeting state-of-the-art DFT references (PBEsol, r$^2$SCAN, and PBE0) to
tackle this problem in all known phases of Li$_3$PS$_4$ ($\alpha$, $\beta$ and
$\gamma$), for large system sizes and timescales. We discuss the physical
origin of the observed superionic behavior of Li$_3$PS$_4$: the activation of
PS$_4$ flipping drives a structural transition to a highly conductive phase,
characterized by an increase of Li-site availability and by a drastic reduction
in the activation energy of Li-ion diffusion. We also rule out any paddle-wheel
effects of PS$_4$ tetrahedra in the superionic phases -- previously claimed to
enhance Li-ion diffusion -- due to the orders-of-magnitude difference between
the rate of PS$_4$ flips and Li-ion hops at all temperatures below melting. We
finally elucidate the role of inter-ionic dynamical correlations in charge
transport, by highlighting the failure of the Nernst-Einstein approximation to
estimate the electrical conductivity. Our results show a strong dependence on
the target DFT reference, with PBE0 yielding the best quantitative agreement
with experimental measurements not only for the electronic band-gap but also
for the electrical conductivity of $\beta$- and $\alpha$-Li$_3$PS$_4$.
