The Propehcy was spread to every single planet. It did not matter what planet they went to Paul was already the chosen one. The only thing arrakis did was commune him with Spice. The whole point of Paul was to be the perfect being, he was born just 1 generation too early according to the Bene Gesserit witches. He was supposed to be born a woman and married to Feyd and their son the perfect being. Now the water of life part is muddied and more so a catalyst like taking steroids but either way he would have eventually unlocked his Prescience.
Dune fans spend too much time debating whether Paul was a hero or a villain while ignoring the woman who helped create the monster.
Jessica knew the Bene Gesserit plans, understood the power of prophecy and still chose to manipulate the Fremen’s beliefs to secure her son’s survival and influence.
Paul inherited the prophecy, but Jessica helped turn it into a weapon.
And here’s the uncomfortable question: Was Paul really the mastermind of Dune’s tragedy, or was he simply the most successful product of his mother’s ambition?