LSH and Arya's similarities are intentional, but LSH is meant to be merciless, while Arya is about learning mercy. Arya's primary motivation is "with Jon to the Wall," the list is secondary. GRRM has been teasing it for a long time. but he always stops her in one way or another. In GRRM's mind, Arya is not ready to go yet. How do I know this?
Hint 1: In the original outline, she is already at the Wall
Hint 2: She was set to go with Yoren, the black brother who had only lost 3 people in his decades career before, yet he dies this time. Such a coincidental death. In GRRM's mind, Arya is not ready to go yet. (Bonus: Weasel and her mom were Arya and Catelyn's plot from the original outline, now changed)
Hint 3: She wanted to go to the Wall before going to Winterfell, but Arya is not ready to go yet:
The kingsroad wound its way past Winterfell on its way to the Wall, and Yoren had promised he’d leave her there with no one any wiser about who she’d been. She yearned to see her mother again, and Robb and Bran and Rickon… but it was Jon Snow she thought of most. She wished somehow they could come to the Wall before Winterfell, so Jon might muss up her hair and call her “little sister.” She’d tell him, “I missed you,” and he’d say it too at the very same moment, the way they always used to say things together. She would have liked that. She would have liked that better than anything
Hint 3: She wanted to go the Wall instead of Riverrun when captured by the Brotherhood without Banners. Preferred being with Jon over Robb (big surprise). Arya is not ready to go yet.
“My lady?” Ned said at last. “You have a baseborn brother… Jon Snow?”
“He’s with the Night’s Watch on the Wall.” Maybe I should go to the Wall instead of Riverrun. Jon wouldn’t care who I killed or whether I brushed my hair.
Hint 4: She asked the captain of the ship to send her to the Wall, he said no. Arya is not ready to go yet.
Her home was gone, her parents dead, and all her brothers slain but Jon Snow on the Wall. That was where she had wanted to go. She told the captain as much, but even the iron coin did not sway him. Arya never seemed to find the places she set out to reach.
Hint 5: She wanted go with Daeron, he didn't want to go back. Arya is not ready to go yet.
When Dareon had first appeared at the Happy Port, Arya had almost asked if he would take her with him back to Eastwatch, until she heard him telling Bethany that he was never going back.
Hint 6: She wanted to ask Sam, but he wasn't going to the Wall. Arya is not ready to go yet.
“You can have the last clams if you want. It’s dark, no one will buy them now. Are you sailing to the Wall?” "To Oldtown."
Hint 7: She keeps telling us she wants to go to Jon. GRRM stabs Jon, the only reason she wants to be at the Wall. Arya is not ready to go yet.
She will have some more filler like this, before finding the motivation to go back to Westeros. My guess about the filler is that it involves Pentos and Lys..