And in both of your cases, the story you are talking about comes from the actions in the game. Like "Hey, remember that time your gnome got mad and bullrushed the duregar off the ledge and into the lava pit, forcing the minions to flee? That was awesome!" or "Remember that tower we took from the mimic king and built a sandwich shop?"
I have those stories in my game too, the emergent ones that only happened because of the way the action happened to shake out.
But there are other, more important, stories in my game that aren't emergent. They are a cooperative tale created by players and GMs that explore deeper things that aren't tied to the mechanics of the game. When my war torn fighter stumbles back home and collapses into the arms of his spose, finally releasing all the anguish he has been bottling up in his stoic turn as the party lead. That's not a challenge. There werent any die rollls, and it's hyper specific to that one individual character.
If that same stoic fighter died to tetanus from a rusty nail on the way home, I lose that thread of story. The family he has at home, the relationships he has with his friends, the rivalries he has built with adversaries....it's all gone.
If I have to create a new character it's like I take an eraser to a giant blackboard of story threads I've developed over time (some emergent from the mechanics of the game and others weaved into the story) and have to start over with a blank slate of New Guys story.
I fully recognize almost everyone's game develops story, but emergent story is not the same as authored story, and authored story is what you can't get from a boardgame and is what I play RPGs for.