Puppet or puppeteer?

In the movie poster and book there is the puppeteer hand holding the title.

Is the Godfather supposed to be the master moving its puppets and controlling the organization?

Because to me, he is the biggest puppet of them all, he is unwillingly pulled into all of this mayhem, never ends up with what he wants, and he is consumed and reduced to a shell of a human by the third movie.

What was the original intention there?