Correct me if I am wrong about this.
But Guardians are basically dead things that were resurrected and sustained by the Traveler's Light, given by their Ghosts.
So...
When Ghaul shackles the Traveler and cuts us off from it, and taking all of our light in the process...how do we survive?
Well, plot armor of course, but besides that.
The Vanguard doesnt necessarily have the heavy duty plot armor, so how do they survive having their light taken away?
Why don't they instantly die?
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Same reason that Eris is still alive. We become drained of light, but it does not sustain us. Only unless we are killed or our light is completely sucked out, like crota's sword.