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Whitebeard was 1 of the Emperors during the era of Gol D. Roger, at this point, we can assume Shiki was also one because he was had the largest fleet. When Roger died, Shiki said to Whitebeard that this age was essentially his as Shiki himself had lost his fleet due to being imprisoned. Roger was dead and Whitebeard was the only great force in the Grand Line capable of ruling.
Almost 2 decades latter, this still holds true; both Akainu and Joker confirm this; Whitebeard is the King that never became King.
First thing that needs pointing out is that Whitebeard was already aware of Ace’s bloodline just as Sengoku said, and those of the upper Echelon within the World Government included; they knew he could potentially be Pirate King being the blood of Roger, so Whitebeard decided to groom him.
Fun Fact/Digression: How did Whitebeard and the Marines deduce Ace was the son of Roger?
Continuing off, Whitebeard stopped trying to become Pirate King because Roger most likely told him about the Will of D, the details of the Void Century and how it would result in bringing down the corrupt World Government;
this is why when Shiki sojourned on Whitebeard’s ship, after Roger’s death, Shiki said it’s Whitebeard’s age, but Whitebeard told him to shut up because he no longer was interested in becoming Pirate King; quite the coincidence.
Whitebeard must have believed Roger’s story and decided becoming Pirate King was not something he needed to pursue any longer. He surely believes this prophecy will come to pass which is why at his death, he proclaims its existence to encourage others to continue pursuing after One Piece to make it both difficult for the World Government in their role as antagonists and to fasciliate to the fulfilment of the Prophecy; the more Pirates go to the Grand Line, the less control the World Government have over the world.
Whitebeard: “Someday, one will arise who will challenge the world.. and shoulder centuries of history. Sengoku, you and the World Government fear the great war that will come to this world one day! I’m not interested in the treasure, but when it’s found… the whole would will be turned upside down! Someone will find it. That day will come.”
Sengoku has an idea of the implications of what searching for One Piece means and the result is another age of Piracy or a continuation of the age of Piracy.
If Whitebeard weren’t influenced by Roger’s story, he would have become Pirate King after Roger’s death and would probably have killed Ace too given the chance. I believe it’s because he knew the significance of One Piece that he decided to rear Ace up to be Pirate King rather than to hate him for being the son of his enemy like so many people did; we can also see he did the same on behalf of Luffy because he saw he had the potential to be the one Roger was speaking of in this prophecy spoken and more so since Ace died.
*Theory by HPsyche