The metal does indeed reduce the napthalene to the radical anion, but that is more just to "solubilize" the electron. Usually Birch reductions are done in NH3 solvent, which can do the solvation by itself, but in more organic solvents like THF, the napthalene helps the electron along.
You idea about the pi-pi interactions is a good one, and is kind of close to reality, probably. I think of the napthalene as more of a shuttle for the electron though: first it receives the e- via SET, and then delivers the e- via another SET to the substrate.