I don't think it's about arresting the decline of China, per se. There were huge structural problems that faced the late Qing, and even a superb government would not have been able to stop those processes from working their effects. Moreover, this was also a relative decline. In the time it took for an able government to carry out basic reforms and secure its rule, the European powers (and America, and Japan etc) would have moved further ahead.
Nor do I think that Zeng would only have been able to take power in the event of a worse rebellion. He probably had the capacity to take the throne in the early 1860s, and would probably have secured the support of the other generals (like Li Hongzhang, for example) who would have represented the only serious threat to his army. I really think that Zeng
chose not to take power, rather than lacked the opportunity.
A better question is to what extent a new dynasty would have been able to a: slow that relative decline, and b: ensure that when the state hits its relative nadir, its better off than OTL.
I think that one big advantage will simply be the fact that the new dynasty will have the excuse to cut out the dead wood. The Meiji reformers were not actually revolutionaries by temperament (indeed, many of them began as reactionaries.) But because the Shogunate was so thoroughly swept away, the new government could install its leaders and ways of doing things at every level of the administration, rather than being a prisoner of existing structures. There were many able Qing reformers- Prince Gong, Li, Zeng himself and even at times Cixi. But the weight of court politics, of entrenched systems was a serious problem.
It's important to remember that the scale of Qing decline wasn't always apparent at the time- right up until 1895, it was an open question in British policy circles about who would make the better partner, Tokyo or Beijing. I actually think that in many ways it was the First Sino-Japanese War, even more than the Boxer Rebellion that really did the critical damage to the Qing. And that defeat, or rather a defeat on that scale, could be avoided relatively easily.
Assume, just for a moment, that we still see a war with Japan on schedule. It would obviously change quite rapidly, since Korea would probably alter in fascinating ways with the Qing falling in the 1860s, but let's just go with it for now.
Zeng's dynasty, with greater control of administrative appointments is not going to see the same systemic misdirection of funds that so crippled the late Qing military. It's possible, likely even, that they still have the problem of their being multiple fleets without a working chain of command, though I do think Zeng's credentials as a military man and reformer will alleviate this slightly. But the fundamental crisis of supply is unlikely to happen- we can expect the Chinese to enter the war with sufficient ammunition for its ships and a half-decent training regime. Even if the Japanese still sweep the Chinese from the sea, it's likely that they don't enjoy anything like their historic success on land. In fact, it's even possible that a Chinese government dominated by reforming generals with more experience of modern warfare (against Turkestan, against France, against the Taiping) will create an army with cohesive doctrine and good enough logistics to win in Korea.
If they do, defeat at sea will be forgiven- the new dynasty will have demonstrated that it can defend China's interests in its traditional sphere of influence.
All of this glosses over the severe industrial, economic and social problems that will face China. But I would actually point to the current work of
@heraclius with his series on the mid-to-late Western Roman Empire. This could be an era in which China, as in our timeline, experiences crisis after crisis, shock after shock. It would still be a time when the oldest of orthodoxies is challenged, in which the Imperial government struggles to stamp its authorities on the provinces, in which foreign powers dominate huge swathes of the economy.
But from our perspective, safely ensconced in the knowledge of Xinhai, the Warlord Era, Chiang, Mao, and Unit 731- it will seem a story of resilience and survival.
