Ok, I actually read through all 14 pages (100pp), so I don't think I'm retreading over too much territory. (I'm sorry this is long, and I'll warn you that these are only my theories even though I believe they fit.)
I'm surprised there hasn't been much discussion about the princes and the Lothric bloodline in general because I found them pretty fascinating. I’m of the school that thinks the queen of Lothric is Gywynevere (this is important to a theory I’ll expand upon later) with that being damn near confirmed when you compare the descriptions of Divine Blessing from DS1 and DS3.
DS1: “Holy Water from the Goddess Gwynevere. Fully restore HP and undo irregularities.
The Goddess of Sunlight, Gwynevere, daughter of the great Lord of Sunlight Gwyn, is cherished by all as a symbol of bounty and fertility.”
DS3: “Holy water blessed by the Queen of Lothric.
Fully restores HP and cures ailments.
The Queen of Lothric, married to the former King Oceiros, was initially revered as a goddess of fertility and bounty. After giving birth to Ocelotte, her youngest, she quietly disappeared.”
We know too from the cinder that drops from the princes that the Lothric bloodline was “obsessed” with producing a worthy heir. If you read all the items relating to Lorian, you get the idea that he was a pretty powerful knight capable of beating a demon prince. Looking at his physical form during the twin prince’s boss fight, you can see that if he wasn’t crippled by his brother’s “curse” he’d be a physically imposing man. Contrast that with items relating to Lothric that describe him as, “destined to be a Lord of Cinder…despite being born into illness, a frail and shriveled child.” (the armor of prayer set) Why was it that Lothric, despite being the younger brother AND sickly, was the heir to the Lothric bloodline and expected to be a Lord of Cinder?
The rest is speculation that I think fits with the information available.
My first theory is that the “heir” the Lothric bloodline was obsessed with producing was actually an heir to the first flame or possibly Gwyn. I believe they wanted to take the power of the First Flame for themselves (their bloodline basically) much like the alternate to the end of fire ending where you kill the firekeeper when she removes the flame. Oceiros and the Lothric Kingdom had control of the Archives which are heavily implied to be Seath’s archives. He could have learned of Gwyn, the first flame, and the power of the lord souls from the knowledge there-in.
My second theory is that Lothric and Lorian are half-brothers. Lorian was around doing his thing slaying demons when a literal goddess appeared. Oceiros, again obsessed with creating a worthy “heir,” has a child of Gwyn, the first lord, at his fingertips and marries her. This to me is a possible explanation as to why Lorian is basically shunted aside despite being older and stronger than Lothric. The only problem is that Lothric is born very frail. I believe Lothric’s frailty is also the “curse” that is spoken of. There are so many expectations placed upon Lothric to be this literal (thanks to his mother, Gwynevere) deific champion that when he comes out weak and frail it’s seen as a curse. Being a dutiful son, I think Lorian somehow fused his soul to Lothric’s possibly to extend Lothric’s life.
We know Lothric is a Lord of Cinder, but how in the world can someone as frail as he is claim the power of the First Flame let alone link it? My last theory for now is that I think Lorian, unable to die because his soul is fused with Lothric’s, linked the flame in Lothric's stead. If you notice all the other Lords of Cinder are basically bestial (including Lorian), but Lothric is quite lucid. If I recall, linking the flame basically causes you to go hollow. However, Lorian can't die because his soul is linked with Lothric's so they have nothing to lose (so to speak) using him to attempt to take the power of the First Flame. If you notice the inventory picture used for the Soul of the Twin Princes, the upper left portion of it is gray--ashen. I believe this is due to Lorian perhaps being incinerated but unable to die.
I believe it’s at this point, after Lothric and Lorian's failure to usurp the flame, that Oceiros started expirementing with the dragon fusions and creating Ocelotte. I think Gwynevere stuck it out for this one last child and saw that this whole thing was madness and then nope’d out.
Please correct me if anything is too out there. I'll also write more when I think of it, because as I said, I find the Lothric bloodline interesting in this game.