The polarization $Q$ gives rise to a positive definite form $Q'$. Automorphisms preserve $Q'$, so they lie in a compact group. On the other hand, automorphisms preserve the lattice, so they lie in a discrete subgroup of a compact group. This gives finiteness. In the case of a Riemann surface, the polarization comes from the cup product, so it's canonical. Putting  together with with your remarks gives finiteness for the automorphism group for a compact hyperbolic curve. As far as I can tell, this argument doesn't the Hurwitz bound of $84(g-1)$, but it works in other situations.