You cannot take it so literally as the actions of only one person that matter. Rand has set in motion dozens of courses of action that probably end with a Prophesy fulfilled, characters have spoken about how the Prophesies are so cryptic he could have fulfilled dozens without them even knowing it. If Rand does not save existence then by default nothing else can happen.

And YES by saving existence itself Rand is indirectly responsible for anything that happens after. Clearly we see potential for people to make their own choices, but without Rand those choices are not possible and thus trace back to Rand. This does not mean Rand must do these deeds himself in order for Prophesy to be fulfilled, his mere existence enables the Prophesies to unfold.

His personal actions are almost incidental to the fulfillment of macro-level Prophesy such as the death of almost an entire race of people, unless you believe he will personally kill all but a remnant of the Aiel. His choices (i.e. saving those who,will accept the Peace of the Dragon) set Prophesy in motion... Thus twenty years after his death/dissapearance the Aiel who did not follow his Peace choose to wage an unwinnable war and end up dying out. They no longer have a purpose in the Pattern and are thus cut from it. Could as easily be some kind of plague or they could in fact die during TG. There are many options for the vast majority of the Aiel to die... Aviendha's vision is simply one of those options. An unlikely one as it clearly shows the Aiel in flagrant disregard to their own Prophesy, something they do not seem likely to do.

Think of how the Mirror Worlds work, they split off into infinite options based on an individual's choices/actions in life. Would Nynaeve of the main world be responsible for the actions created by the Nynaeve of a Mirror World? Or the actions of a Nynaeve of a Mirror World several times removed from the real world?