  Observations of GW170817 strongly suggest that binary neutron star (BNS)
mergers can produce rapid neutron-capture nucleosynthesis (r-process) elements.
However, it remains an open question whether BNS mergers can account for all
the r-process element enrichment in the Milky Way's history. Here we
demonstrate that a BNS population model informed by multimessenger neutron star
observations predicts a merger rate and per-event r-process element yield
consistent with geophysical and astrophysical abundance constraints. If BNS
mergers are to explain the r-process enrichment of stars in the Galaxy, we
further show using a one-zone Galactic chemical evolution model that they have
to merge shortly after the formation of their progenitors, with a delay time
distribution of power-law index $\alpha\leq -2.0$ and minimum delay time
$t_{\rm min}\leq 40$ Myr at 90% confidence.Such short delay times are in
tension with those predicted by standard BNS formation models and those
observationally inferred from samples of short gamma-ray bursts (sGRBs).
However, we find that a two-channel enrichment scenario, where the second
channel follows the star formation history, can account for both Galactic
stellar and sGRB observations. Our results suggest that 45-90% of the r-process
abundance in the Milky Way today was produced by a star-formation-tracking
channel, rather than BNS mergers with significant delay times.
