You do what she wants either way- you get the lord soul to play into Gwyn's fake enslaved cycle, or you also do it as the fool she sees you as to kill off her sisters who she betrayed Gwyn's faction too because she didn't quite like the idea of becoming a Demon. Either way, Izalith dies and the Demons slowly die out as a result.
The goal is usurp the flame and end the egg princesses illusion, but everyone left shit for too long- which is why you're left in a barren dead world at the end of DS3, the first Idiot linked the fire and started the fire-linking meme, which Aldia figured the bullshit out and a while later the Fission-Spawned THOT offpsring of Manuses Dark soul gone wild showed up and ruined almost every kingdom in the area, except the ones who got hot dickings. Then Nashandra tried to Usurp the flame and was stopped by an undead that found a way out and took it, then we move to DS3 where there's been shitloads of fire-linkers and the supression of mankind, giant-kind etc had hit an all time high of slave-morality to the max.
Then a nutty guy from a painting shows up, find a nutter born from a blind mother who can do divinations of the future and debases himself by cannibalism and sets him on Anor Londo so painting boy can rule unattested, but you kill them both, (And send Alritch where he wants to be with his trap god powers to boot, at the cost of his soul.) And some stuff happens such as the snakes snuffing it, the sable church being formed and their one attempt to Usurp the flame failing, so they sort of wait it out for you to show up, what with them already winning the support of the other guy who was going to link the flame.
Anyway, you get the Lords of Cinder, beat the Soul of Cinder, and at the crucial point, eat the first flame and Usurp it, slowly allowing the world to progress in it's natural state- also relating to the eyes of the first firekeeper, who is none other than egg girl, which you giveto your firekeeper to show her the truth of her nature, and some other stuff.
Anyhow, Ringed city- and shit's over.
Except Aldritch, rendered unto the earth isn't digested, or at least is hard to chew- and he makes it to that tree-ridden layer, of the world full of water, avoiding total ego death, though he no longer has his lordsoul NOT THAT THAT FUCKING MATTERS, BECAUSE AFTER THE TIME SKIP EVERYTHING HAS SUNKEN UNDER THE EARTH, NOW, LETS EAT THE PROFANED FLAME, ANYTHING ELSE RELEVANT STRAY UNDEAD, AND ANYTHING ELSE THAT COMES HIS JOLLY WAY, Oh right, and he still has the powers of illusion and everything Gwyndolin has, and this, technically a Son of Gwyns powers.
Cut a few eons later, and he's the Old one from Demons Souls, Evident as the extension of his body is the maiden in black, made in the likeness of his own mother, (Blind, wax on eyes!) the soulless ones are called DREGLINGS And he himself, turns Allant into a Sludgelike amorphrus monster- Gee, where has THAT been seen before? Oh, right, the fact that he's a giant tree, which given his nature as an undead means like all others, his humanity was drawn out into a tree like the ones you see literred everywhere, except his dear Dregs kept him landlocked, so he literally made himself a bark shell. Oh, and he's in a meaningless ammoral age of men where nothing has any meaning whatsoever, and he is also eternally hungry because he has no soul, and instead self-cannibalizes his own offspring and the souls of men produced from his being to circulate a functional ecology.
Oh and again, he's at the very base of the earth, and is immortal and cannot be killed. Because boneless Undead God blob born from a what, now? A Demigod? Did I forget to mention he's defeated by being 'put to sleep?' Wait, shit! that's exactly what was done with Aldritch the first time! and he Envisioned his age, eons into the future!
Did I forget to mention that the fucking things stuck to the Old Ones back ARE THE FUCKING PROJECTILES ALDRITCH USES WHEN HE WAS STUCK TO GWYNDOLINS UPPER HALF? OR THE RUINS OF CIVILIZATION ON HIS LEVEL? At some point, during the growth of bark on his body due to excess humanity, the dear may have tried to pry his new 'shell' off! Nah, couldn't be.
Didn't make the deadline.