Comment by JoshTriplett

It's possible to have a reasonable story even then, with two stipulations.

First, obviously, that L is at least as good as Light; you can't scale up one side and not the other or you don't have a story.

And second, that Light fundamentally wants to be perceived as a real force, since he's nominally trying to deter crime. Sure, if he acted slowly and randomly, he could appear nearly indistinguishable from random chance, but he wants to send a message. That stipulation is enough to force him to reveal some information, and combined with a sufficiently good L, you'd still have a story.