Personally that's kind of missing the point of what drove him to the mountain, and what he had to come to peace with, at the end of TGS.
It wasn't "I'm going to die" it was "I'm going to die... And it wont change anything. I'll just be reborn to do it all over again, and again, and again.".
Honestly, Rand was one of the few, if not only, Third Ager who actual seemed to "get" what "The Lord Dragon Reborn" actually meant.
For everyone else their was at least a "chance" of a "happily ever after", and that their soul might be reborn in more peaceful times, but for Rand? It's always going to be "The End Of The World".
...Honestly, I always found it odd how no one seemed to get that when talking to him about TG. Like they thought the whole "born to fight the Dark One" was just a two time job for him.
Of course, turns out that the Pattern being all about "balance" means that every once in awhile, it's Chosen One gets turned into an immortal* God-like being.
*At least based on the idea that Nakomi is the same, and how old she'd have to be to be so. Though of course will die at some point, as this is "the Wheel of Time" after all. And man is that likely to be one hell of a LB.