Right
卒 supplies the sound (zú → suì, a regular dental-sibilant alternation) and a faint semantic flavour:
卒 means 'finish, end, die' - a thing reduced to fragments has reached its end. Phonetic family:
粹 cuì (pure),
醉 zuì (drunk),
翠 cuì (jade-green) - all on
卒, all phono-semantic.