Why would conciliation be better than shuttle diplomacy for a family at odds over how to care for the affairs and health of their aging father?

A conciliator will pay attention to the emotions involved in a sensitive family issue like this.

Conciliation is a more streamlined process and therefore a better fit for this situation, which requires a speedy resolution.

Conciliation allows families to have a decision made for them to ease their tensions, whereas shuttle diplomacy requires them to make their own decision.

Shuttle diplomacy requires at least one face$-$to$-$face interaction, which is not possible for many families who live far apart from one another.