>>714655643This
The choice to not link the flame just kicks the choice down the road.
If you kill everyone and leave, someone else will come after you and do it.
Dark Souls 2 is fundamentally about peeling this back. The legend, the myth, the duty, the compulsion for undead to collect souls, to follow the footsteps of the chosen undead, to keep the curse at bay, is a myth, told so undead will continue the cycle. The Monarch ending of Dark Souls 2 is recognizing the quest laid out before you will not break the curse, and so, with souls and dignity, you leave to (presumably) succeed.
Dark Souls 3's firekeeper ending, ends the cycle. The vision of the firekeepers, was the possibility of their betrayal of the task set before the chosen undead across ages. The understanding and realization, that they can, as something of a manifestation of the bonfire, absorb the flame, preventing it from ever kindling again.
What follows is anyone's guess, but the cycle is now over. Nobody will ever be able to link the flame again. No more chosen undead, or ashen ones will be able to complete the quest, to recycle the flame. The fire fades, and true Dark sets in, either forever, or until something new happens.
The Lord of Hollows ending is... sorta the same, except, the curse of undeath remains. Under the fire fading ending, presumably, mortality takes the world, and everything is allowed to finally die. The age of hollows, undeath remains, and hollows continue to exist for all eternity. It's a bit unclear if the higher forms of the curse remain, the derangement and so on, but, undead now exist through the endless, formless Dark that follows.
And, in a bit of a reversal, this, too, is now inevitable. Just as before, someone would always come along to eventually link the flame, now, it's only a matter of time until an age comes where none remain who can link the flame, or another comes to end it. The only real choice you have, is if Humanity fades too, or persists through undeath.