While true, I would counter that there is an imbalance in the potency of energy sources involved here.
The emperor binds licensed psykers to himself in much the same way that chaos gods bind demons. He can "eat" them at will. (and eats many of them daily to sustain himself.) He is able to tap enough raw energy to fire a beam of coherent immaterium all the way out into intergalactic space.
The chaos gods are able to sustain the eye of chaos.
A good portion of the psykers that are bound to him in this fashion are bound willingly. (probably because they have seen what happens to unbound psykers when hungry warp polyps take notice of them, and the idea is fucking scary as hell.) Fear of the warp, and faith in the protection of the emperor, are two sides of the same coin. One produces and sustains the other.
The gods do not sustain the Eye of Terror, it is self sustaining drawing from the near infinite energies* of the Warp not controlled by the Gods.
The Emperor also cannot consume sanctioned psykers at will and is only soul bound to Astropaths, which he also cannot consume at will. He is fed the dregs of the psyker tithe through a mechanism in the bowels of the Golden Throne that consumes their souls and uses it to feed him and power the Astronomicon. Without the Throne he can't consume any souls at all and would probably just die instantly.
The Astronomicon is something he's sustained since before being worshipped, it's just a giant beacon of undirected psychic energy radiating from Terra. Without the Astropathic Choir that directs and focuses it for him it'd have substantially less range and it's gradually getting weaker as the Golden Throne slowly breaks down anyway.
He's not being powered by worship, he can just manifest some blessings upon and through the faithful, he's powered by shoveling low grade psykers into a metaphysical furnace like coal into a steam engine in the thousands per day, and because he's still dying it's getting harder and harder to keep him going. He can't even communicate with people who aren't in the throne room and able to physically see him.
*All warp entities have theoretically infinite energy at all times because time is weird in the Warp as is space, but it's never directed in any given place and doing so would probably invite bad stuff for whoever tried to do it, chaos god or otherwise.