If Paleblood is the Moon Presence, why do you ask the doctor in the intro about it? You wouldn't know about it until later, no? That part confuses me but the rest seems very sound.
There's a note in the clinic at the beginning of the game that says, "Seek Paleblood to transcend the hunt." Since you don't know anything about it, you ask Gilbert, the first friendly NPC, who says he doesn't know anything about Paleblood either, but points you toward the Grand Cathedral because it's the home of Blood Ministration and they have many kinds of unique blood. That starts your journey.
In the Lecture Building, a note states, "The nameless moon presence beckoned by Laurence and his associates. Paleblood." That's the strongest association together, but it could also mean that Paleblood is what beckons the Moon Presence, who otherwise has no name. In this case, the PC is Paleblood, sought out specifically to engage the Moon.
The third of Cord's flavor text (Abandoned Workshop) reads "Every Great One loses its child, and then yearns for a surrogate. The Third Umbilical Cord precipitated the encounter with the pale moon, which beckoned the hunters and conceived the hunter's dream." The others all have a similar vein.
Arianna's: "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."
Mergo's Wet Nurse: "Every Great One loses its child, and then yearns for a surrogate. This Cord granted Mensis audience with Mergo, but resulted in the stillbirth of their brains."
Iosefka: "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."
The conclusion is that there are multiple Outer Elder Gods, and this is their means of reproduction. Arianna's child is Oedon's, and Yharnam's is Mergo's, but was stillborn. It also suggests that finding and using a Cord also is a way to communicate directly with that God (Pale Moon; Iosefka's God is unclear, but that was the function).
