quietly, softly, solemnly
肅 centers on a restrained, dignified stillness: acting quietly, softly, or with solemn respect. The three given meanings all point to a subdued, serious manner.
肅 is traditionally explained as a hand holding a brush over a vessel or frame, suggesting careful, solemn action. The exact historical development is uncertain, but the character has long been associated with respectful, quiet behavior.
The top part looks like a hand holding a brush, and the bottom resembles a container or frame. Imagine a scribe carefully writing a solemn document in complete silence: every stroke is quiet, soft, and serious.
For シュク, picture a solemn monk in a quiet temple, gently shaking a shaku (a ritual wooden scepter) to maintain the hushed atmosphere: shaku -> シュク, and the soft sound fits the solemn mood.