Well actually they aren't saying "now it's finally China's turn", they are saying "after 200 years of temporary embarrassment China is returning to its rightful natural position as hegemon."

Thing is China was never hegemon of anywhere but East Asia, and really not much more beyond the lands that currently are part of China, like Xinjiang and Tibet. It never had hegemony over India, and in fact you see a lot more Indian influence on Chinese culture (Daoism, Buddhism, those statues of lions everywhere when China has never had such animals) and Islam spread to China but Confucianism never spread to Iran and the Stans. Not to mention Europe, Africa, Oceania, and the Americas were mostly unaware of China's existence, and China was unaware of them. Even by the 1800s Chinese officials believed Europe had a population of 10,000 and could be defeated by cutting off their supply of rhubarb. The glorious history is largely self-delusion and propaganda.