So, each team member has an ability to show what they give to the team right? Is that all it does, because it is worded like they can use that power. If that is the case, then would they be granted that power like a trait, or would it only work if the asset dial was showing that symbol?

Yes, that's all it does. The individual team member (if played on another team, or played via Solo Adventure) does not have access to their asset dial ability because...well, they have no asset dial. That ability only shows what the team base gains by having that member attached to it.