Okay, I'll take a crack at this.
Okay, first off if we're talking eldritch horrorbirds we've got to be talking hueg. So, let's make this thing a good 15 feet tall, probably something like 20 or so feet long tip-to-tip. Now that it's hueg, time to make it fucking scary as hell. Keep the giant unformed eyes, the shiny, flesh-coated beak, and the general pinkness of the whole thing. Also, it has four heads each extending from the main body on a long, thin, and weak looking neck, craned in odd twisted forms. The wings will be more akin to shattered arms, with two digits at the ends for manipulation, the rest extending back like normal wing-bones.
Now that it at least looks scary, it has to have some alien motivations to be a true eldritch horror.
These creatures cannot see the world that they depend upon to survive, so let's roll with that.
They are unnaturally hungry and needy. They are in constant need to be fed and cared for, so they're likely pathetically weak. This need is something I can run with, though. You see, each of their gaping beaks constantly is open in that 'dear god, feed me' look baby birds always have, and they constantly speak in whispered tones. They constantly ask of their mother, screech for sustenance and love, and know your deepest hidden sins. These monstrosities feed upon your emotions and will extract every last drop of your spiritual essence.