  We investigate the dynamical properties of Sr$_2$RuO$_4$ at zero and very low
temperature using density functional theory plus dynamical mean-field theory
with an exact diagonalization solver. By considering rotationally invariant
local interaction, we examine how Hund's coupling and spin-orbit coupling
affect the correlated nature of the system. In the absence of Hund's coupling,
the system shows a Fermi liquid behavior over the entire range of temperatures
we consider. We confirm that the Fermi liquid persists at zero temperature even
with nonzero Hund's coupling; however, at sufficient temperatures Hund's
coupling significantly reduces the Fermi liquid regime and the system evolves
into a typical Hund's metal. At the bare electronic occupancy of Sr$_2$RuO$_4$
($t_{2g}^4$), a stronger Hund's metallicity accompanies a larger long-time
correlator. Remarkably, electron doping further destabilizes the Fermi liquid
even though the long-time correlator and magnetic fluctuations decrease upon
doping. This suppression of the Fermi liquid is driven by the van Hove
singularity above the Fermi level in Sr$_2$RuO$_4$, combined with an enhanced
Van Vleck susceptibility by spin-orbit coupling. Such findings point to the
important role that electronic structure plays in the behavior of Hund's
metals, in addition to magnetic fluctuations.
