In formulating my example upthread, I had in mind a more complex example that I omitted at that time, which was to compare with Charisma (Persuasion). It's worth laying out that example now.
...(snip)...
Across D&D groups I have seen DMs grant anything from healing to sanctuary to information to escape routes as a result of Charisma (Persuasion) rolls. With NPCs, it is often the case that the group do not know what parts they may go on to play. A roll to influence can determine that. For example, the priest above could just as well have been persuaded to heal, provide sanctuary, quench thirst or satiate hunger, illuminate the area, etc.