Yet these foreigners, especially the Qing, adopted the Chinese Imperial system and over time basically became Chinese, so much so that the Manchus lost their language. The Yuan didn't adopt the Chinese system and were kicked out by a peasant rebellion which then became the Ming. As for the Ming, I wouldn't consider them one of the greatest Chinese dynasties, but they kept the ball rolling from the 14th to the 17th centuries.
I think its wrong to consider that each of the Chinese dynasties separated from each other in the manner you arrange it in, but rather as simply a part of the cycle that represents Chinese civilization as a whole.
Therefore, to answer your question, the Asian Empire that has been by far the most successful has been China, plain and simple. I could go on to say, 'they were the first to introduce this... they were the first to explore that...' etc, but they managed to create a political system that lasted for thousands of years. When one dynasty fell, another would rise to take its place, adopting the foundation of previous dynasties.
Could China have been stronger? Of course. We could argue that Zheng He could have gone further, reached Europe and made the Pope and all the European monarchs of the 15th century the Chinese Emperor's bitch. The reason why the voyages didn't continue is like what happened with NASA, domestic concerns and more immediate foreign policy concerns (like the Mongols) took priority and the funds dried up. As for the 19th century and the first half of the 20th century, they certainly could have gone better for China. And of course there were nomadic invasions like the Mongols and the various phases of disunity that always seemed to plague China every couple hundred years.
As for other Empires, I think the Ottoman Empire could have survived had there been reform sooner or even with a better outcome in WWI, and thus we wouldn't have the Sadaams, Ahmadenejads and Osamas of OTL, there wouldn't be Wahabbism or other extreme fundamentalist ideals (that's not to say there won't be other problems but I can only speculate).
Potentially, had Japan not retreated into isolationism during the Tokugawa but rather continued trade and technological development, they could have become like Asia's Great Britain, with a powerful navy that could police a sizable colonial empire. Still, considering that in the 19th century they could have become a colony they did alright, save for WWII of course...
Well, if you want to be technical, let's do that list again, with modifications.
1) It was founded in China.
2) It was founded by a Chinese ruling class.
3) Most of its population was Chinese.
4) Most of its successor states were located somewhere in China.
5) Most of its territory was in China.
6) Most of its number of languages spoken was Chinese.
7) At least one of its rulers had Chinese heritage.
8) At least one of its rulers was fluent in a Chinese language.
9) It had territory that was considered culturally Chinese.
The Mongols only fulfill requirements 3 and 9.
The Qing only fulfills requirements 3, 4, 5, and 9.
So neither were Chinese dynasties, meaning that China remained strong until the 1400's. Besides, the Manchu were never truly integrated into China until the 1800's, which meant that they remained independent for about 2000 years. In fact, there was no substantial population in Manchuria until about that time.
I'm not saying that China wasn't the strongest country in the world, although you could easily make arguments against it. For example, I highly doubt that the Chinese could have been any stronger. The Ming simply would not have been able to travel to Europe, because they were mired in ethnocentrism, which meant that they had absolutely nothing to learn from 'barbarian' civilzations, so it would be a complete waste of time to travel around the world trying to establish colonies. Remember, the Chinese had no interest whatsoever in trying to 'civilize' the 'barbarians,' like the Europeans did.
As for the Ottomans, I doubt that they could have been any more stronger. By 1700-1800, there were various ethnic groups clamoring for independence, most of which succeeded. Eventually, by the end of the 1800's, it was referred to as the "Sick Man of Europe" for good reason. It would be almost impossible to hold Eastern Europe together because of the legacy of the Byzantine Empire, and the Arab world, because the Ottomans were Turkish, not Arab.
Japan couldn't have become that much powerful prior to the late 1800's. They weren't even an empire in the 1500's, when they invaded Korea, and the fact that they were simply unable to defeat a lone general (Yi Sun-shin) with extremely insufficient resources just goes to show you how weak thee Japanese were. I really don't think that the Japanese could have industrialized any faster in the late 1800's, unless you could managed to do so in less than three years, which is simply ASB and just ridiculous.
Now let's add Goguryeo/Balhae to the mix. Both civilizations, especially the first one, managed to take advantage of China's weakness and fragmentation, and managed to dominate a significant part of Manchuria. If they had tried any harder by setting their sights on North and South China, then they would have probably been capable of uniting China and managing to establish a firm and stable presence in the Korean Peninsula, Japan, Manchuria, and China. Look at my timeline for more details.
The Song dynasty was an early peak in Chinese civilization. The fragile political situation had more to do with society continually out-pacing evolution in the political system, rather than some inherent weakness of the Song era civilization. The Ming were a recovery from the absolute disaster of the Yuan years, and eventually reached and surpassed the Song height.
Seriously, you can't just read a wiki history page to assess the efficacy of one of the Chinese dynasties. Plus, China didn't peak in the 1400's, they peaked in the late 1700's.
If you're going to include the Han, Tang, and Qing dynasties, there's literally no reason not to include the Song and Ming. The Ming are probably the best candidate behind the Qing for 'strongest' Asian empire.
Look at my list in response to your statements that the Qing was a Chinese dynasty.
I'm not saying that neither the Song and the Ming were strong; in fact, I agree wholeheartedly that both managed to restore Chinese glory after tumultuous eras, and that they were strong as they were in OTL, but both couldn't have been any more greater/stronger because of nomadic invasion and ethnocentrism, which forced them to turn inward.
In terms of cultural accomplishments, look at how modern-day Chinese refer to themselves. Most call themselves people of the Han and Tang, but I've yet to encounter someone who calls him/herself of the Song or the Ming. In fact, we call Chinese the
Han Chinese, further demonstrating the influence it still retains after hundreds of years.