I've seen a lot of posts about running a new wire from the fuse box to the HEI terminal. Forgive my ignorance, but would it be possible to run that wire from the ignition switch of the starter to the distributor?

You could run a wire from the ignition switch (not OF the starter) BUT that wire is already there, it comes from the ignition switch to the neutral safety switch, out of the neutral safety switch to the bulkhead connector.

The only part that needs to be replaced for the HEI conversion is the resistive wire from the engine compartment side of the bulkhead connector to the coil/distributor.