If you read the descriptions of the umbilical cords they mention

"A great relic, also known as the Cord of the Eye. Every infant Great One has this precursor to the umbilical cord."

, but

"Every Great One loses its child, and then yearns for a surrogate."

Most of the cords mention the Moon Presence in some capacity:

The Third Umbilical Cord precipitated the encounter with the pale moon, which beckoned the hunters and conceived the hunter's dream.

Every Great One loses its child, and then yearns for a surrogate, and Oedon, the formless Great One, is no different. To think, it was corrupted blood that began this eldritch liaison.

This Cord granted Mensis audience with Mergo, but resulted in the stillbirth of their brains.

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.

And then there's the note in the Lecture Building:

"The nameless moon presence beckoned by Laurence and his associates. Paleblood."

So Mensis were trying to collaborate with Moon Presence to birth a Great One to evolve humanity into its next stage. This is backed up by a few pieces of lore:

Master Willem was right. Evolution without courage will be the ruin of our race.

and from the trophy's description

You became an infant Great One, lifting humanity into its next childhood.

I'm not sure if Mergo, the baby at the tower's peak in Nightmare of Mensis, was the surrogate for Moon Presence or if it was part of the sacrifice required to summon it, but after defeating the Wet Nurse the ritual is defeated and you're free to awaken from the dream having averted humanity's evolution. When you refuse to accept your death at the hands of Gehrman he questions what it is that forced your change of heart. If you don't consume the cords then I'm guessing it's because you want to take his place as the number one hunter, but if you do consume the cords then you're presumably trying to complete the ritual Mensis started. Why you don't let Mensis complete the ritual is a mystery. Either you didn't realise what they were doing until after you ended their ritual or you knew they were going to fail with only one cord. I don't know. There's a note in the lecture building that mentions three cords may be necessary for the birth, but you only get one from the Wet Nurse, so maybe they were on the road to failure. I DUNNO, MAN!

Anywho, I'm not going to claim to be the expert on this, so everybody else should absolutely jump in with ideas!