Director = Their new God?

We don't know that it's that simple. The Faction may have a completely different set of goals. For example, maybe they have a charismatic leader (Hitler wannabe) and their goal is to save the world and create a new order under them.

The Director isn't their God, but it is a superior intelligence. Over time a benign superhuman entity would win peoples' confidence with its ability plan out and execute complicated tasks (like the management of society's global resources) better than any human being and always making the right choices. After a century or two of consistent perfection, and people challenging its conclusions always being wrong, everyone would tend to give it the benefit of the doubt. Any problem you think you see has probably already been thought of and solved by the Director.

The same thing would happen if a human leader made one uncanny perfect choice after another and never seemed to get anything wrong. People would be willing to defer to their judgment. Nothing mysterious or religious about it. Just common sense.