Plus, the New Era Bellamy talked about is not the same idea Doflamingo hinted at.
Bellamy thought the next age would be for the strong pirates who didn’t waste time with dreams.
Doflamingo clearly didn’t care about Bellamy’s interpretation and talked about a post-Whitebeard era when war would engulf the world. He strongly believed in “might makes right”.
In Sabaody, Dofla talked about the next era being “of Smiles”. From context we now know he was producing artificial DFs so Kaido could start his dreamed world-engulfing war.
Doflamingo’s capture made the era he envisioned impossible, but that meant the world would fall into chaos. “A throne war” for supremacy between Emperors, Warlords, World Government, New Generation and the Revolutionaries.
The “Throne War” began in two fronts: the Reverie, with the abolition of the Warlords, and Wano, with the Emperors being challenged by the New Generation.