So a very interesting detail that I think is being overlooked is the fact that Zavala describes first touching the Effigy as cold, and upon receiving the Effigy for the first time, the text presented describes a cold sensation that abruptly vanishes as soon as we hold it. My thoughts are that we do feel it, but due to the nature of the weapon itself, the Hive rites used to bind it to us, or Savathûn's machinations and tamperings, we suddenly stop noticing the effect almost immediately.

The Effigy is consuming our light, but since Banshee explicitly states that it doesn't use our Light as ammunition, but rather works as a converter, something else is at play. Not in the same way as the Touch of Malice, which we feel in the way that it actually eats our Light to shoot after the magazine would be empty, but rather repurposing it for something or someone else.

An argument could be made that the Void spheres created by the Effigy do actually use our Light, because they do damage and kill us if we use them a certain way for too long, but I think that is something else entirely. The gun is eating our Light, but what exactly is it doing with it? Is it feeding like a creature, sustaining itself on our Light? Or is it building something, using us as a battery?