The cords definitely do something.
Iosefka's one says "Provost Willem sought the Cord in order to elevate his being and thoughts to those of a Great One, by lining his brain with eyes. The only choice, he knew, if man were to ever match Their greatness."
Which means Master Willem probably obtained his eyes/head tentacles from the cord. So consuming all three make you into a little squid. Dunno how Rom fits with this. Was she also given a bunch of cords?
Hmm, I don't think just consuming the cords make you a great one. I'm not sure how defeating the Moon Presence fits into this, but it seems to.
The "Three Third Cords" note is in the same place as the "The nameless moon presence beckoned by Laurence and his associates. Paleblood" note, so they should be related. Either you need three cords to have audience with the nameless moon presence (this event could be what's called paleblood), or after beckoning the moon presence (and potentially losing Gehrman to it in the process) they figured out they needed three third cords to resist it, or something like that?
EDIT: Actually, couldn't that also fit the "Hunt the Great Ones. Hunt the Great Ones." not into all of this? I never really understood that note, but if they figured it out all the way through, that you need three cords and you need to kill the moon presence to become a Great One, maybe this not is to represent "we must consume three cords and kill a great one in order to evolve."
Also fits the recurring theme of evolution
"Master Willem was right, evolution without courage will be the ruin of our species" (or something like that)
And also the Metamorphosis runes.
EDIT2: Ignore that, makes no sense, you can consume three cords before killing other great ones and that won't give you the ending right there, so the game doesn't support this last point.