Technically China's golden age was the Tang Dynasty, but that's a separate post.

The Han dynasty is popular because it occupies a similar cultural position in Chinese history and imagination as the Roman Empire does. Founded on the backs of the shorter lived Qin Dynasty, it was the dynasty that consolidated and expanded upon Qin holdings to roughly map out the contours of much of modern China (excluding Tibet), ruled the region for four centuries, and set the intellectual and cultural template for future empires, including the Tang (they would constantly look backward for Han precedents to follow or adhere to).

Its collapse due to civil war and rebellion also occupies a similar position in Chinese thought to the Fall of Rome, with some of the best works of Chinese literature set in that era, like the Three Kingdoms.

The Ming and the Qing, comparably in eastern thought, are viewed more questionably because of Ming's status as a troubled Chinese dynasty, and the Qing's status as a foreign dynasty whose weakening threw China off its position as global superpower.

The Ming and the Qing however, are popular in western imagination because it was precisely at those times that peak cultural interaction between east and west occurred. So I suspect the fact you hear more about them is due to western biases than any sort of eastern reluctance.