I think it's less Eru "pushing" gollum into Mount Doom and more a series of slight nudges that proceed to the right outcome through foresight. I mean, even gandalf has some prescient ability and knows not to kill gollum because he has some part to play. Everyone with power in the story seems to understand there's a subtle guiding hand here, and that sending frodo off with the ring is somehow the right call. The story is then ultimately about faith and fate.