thrust, poke, come together, assemble
搶 centers on a forceful hand action: thrusting or poking something, and by extension, things coming together or assembling as if pushed into place.
搶 combines 扌 (hand) with 倉, which likely contributes the sound and may suggest a forceful action. The exact historical development is uncertain.
The hand radical 扌 beside 倉 suggests a hand thrusting or poking into a storehouse, making things come together.
For ソウ, imagine a hand thrusting a saw into a crowded storehouse: saw -> ソウ, and the poke makes everything assemble.
winning on a tile robbed from another player's kong