Does a son's rushing during tawaf and sa'y to catch up with his mother affect the validity of his tawaf and sa'y, and is this considered a distraction that harms the validity of tawaf and sa'y according to the Shafi'is?

Rushing during Tawaf (definition) and Sa'i does not affect their validity. This is not among the diverting factors that the Shafi'is stipulated must be absent, because your son did not intend anything other than Tawaf and Sa'i. A diverting factor is when a participant intends the competition itself without the intention of performing Tawaf and Sa'i.