An obvious obstruction to the proposed topological characterization is the following: Endow $H(D)$ with the discrete topology, call it $\tau$. Then the space $\langle H(D) , \tau\rangle$ has the following properties: $f \mapsto f(z)$ is continuous for every $z$, $\langle H(D), \tau\rangle$ is a topological ring (as in, point-wise addition and point-wise multiplication of functions are continuous), and $\langle H(D), \tau\rangle$ is metrizable. The obvious down-side to this topology is that it is not separable and not natural in the same way but it does offer a counter-example to the statement
If $\tau$ is a (Hausdorff) topology on $H(D)$ for which point-evaluation is continuous, the group and ring operations are continuous, and $H(D)$ is metrizable, then $\tau$ must be the compact-open topology.