  Stellar models predict that lithium (Li) inside a star is destroyed during
the first dredge-up phase, yet 1.2% of red giant stars are Li-rich. We aim to
uncover possible origins of this population, by analysing 1155 Li-rich giants
(A(Li) $\geq$ 1.5) in GALAH DR3. To expose peculiar traits of Li-rich stars, we
construct a reference sample of Li-normal (doppelg\"anger) stars with matched
evolutionary state and fiducial supernova abundances. Comparing Li-rich and
doppelg\"anger spectra reveals systematic differences in the H-$\alpha$ and
Ca-triplet line profiles associated with the velocity broadening measurement.
We also find twice as many Li-rich stars appear to be fast rotators (2% with
$v_\textrm{broad} \gtrsim 20$ km s$^{-1}$) compared to doppelg\"angers. On
average, Li-rich stars have higher abundances than their doppelg\"angers, for a
subset of elements, and Li-rich stars at the base of RGB have higher mean
$s-$process abundances ($\geq 0.05$ dex for Ba, Y, Zr), relative to their
doppelg\"angers. External mass-transfer from intermediate-mass AGB companions
could explain this signature. Additional companion analysis excludes binaries
with mass ratios $\gtrsim$ 0.5 at $\gtrsim$ 7 AU. We also discover that highly
Ba-enriched stars are missing from the Li-rich population, possibly due to
low-mass AGB companions which preclude Li-enrichment. Finally, we confirm a
prevalence of Li-rich stars on the red clump that increases with lithium, which
supports an evolutionary state mechanism for Li-enhancement. Multiple culprits,
including binary spin-up and mass-transfer, are therefore likely mechanisms of
Li-enrichment.
