This isn't about the player planning a story arc. It's about the player expressing wishes though his background and leaving questions.
The best backgrounds leave a good question to be answered. In his backstory one of my PC's wound up in jail because a member of a rival merchant house was found bisected in his room at the inn.
So he left me this question as a GM to answer. Who killed the rival and why? And why try to frame it on him?
Another PC in the same campaign wrote a short story where his master in the slayer order was betrayed by a another slayer and killed and his character was left for dead. So he wants to find the slayer that killed his master. A simple revenge story.
The third PC has parents that were connected to an assassins guild and his parents were poisoned when he was 15, his father died but his mother survived, but the incident left his mother invalid so he has spent his time trying to find some cure for his mother. The same night his parents were poisoned his twin sister ran off and he doesn't know why.
The players aren't planning any story arcs, they are providing me with information that allow me to make build adventure, intrigue and mystery. Most of what we play is about the characters, it starts like a small snowball that just gets bigger and bigger as their actions have consequences.