child, servant, diminutive
崽 centers on a small or dependent person: a child, a servant, or something diminutive. The meanings share the idea of being young, subordinate, or reduced in scale.
崽 combines 山 (mountain) and 思 (think), but the historical reason for this combination is uncertain. It may have originally represented a child or servant through a now-obscure phonetic or semantic link.
Picture a small child (崽) sitting on a mountain (山) deep in thought (思). The child is diminutive compared to the vast mountain, like a servant waiting quietly.
For サイ, imagine a sighing child: sigh -> サイ. The child on the mountain lets out a small sigh, and you remember the reading サイ.