I'm actually quite a fan of branching storylines when done right. I could see the game giving you a choice in Zanarkand in FFX. When Yunalesca asks you to choose who will be the Final Aeon, I could see being able to give any of the party members' names for an answer and "No one" being the last choice. Obviously the latter would continue the story as it is, and the others would go on for a bit (without save points mind you) and then end in a bad end as the cycle of death goes unbroken after Yuna's sacrifice.

Things like that. It would strengthen the feeling of being in a desperate situation with little chance to escape it, which to me at least would strengthen the immersion as well. There's something about going down story paths you choose yourself that makes it feel more connected to you.