Many Emperors were actually intimately involved in both the day to day running of the empire and dealing with various crisis's that popped up. Now keep in mind that the organisational structure varied enormously between the various emperors and dynasties, making any sweeping generalisations pretty useless. But by looking a little more closely at the case of the Qing dynasty (1644-1912), we can see that the emperor was central to the entire structure of the imperial state.

The first task that the emperor had was the overseeing of the bureaucracy itself. A dutiful emperor would ideally constantly and closely examine the reports sent from around the empire to the capital and regularly call meeting with his top officials to discuss the current state of the empire and the activities of its officials. If the actions of an official was called into question (whether corruption, ineptness or over-zealousness), the emperor would mete out judgement and punishment, usually in the form of docking their allowances, but in more severe cases the official might be fired, demoted to a minor post on the fringe of the empire or even executed (keep in mind though that one of the perks of being a graduate of the imperial examinations was that you were immune to torture and could only be executed by strangulation, as opposed to beheading or "slow slicing"). The emperor also possessed ultimate authority in the organisation of the bureaucracy, particularly in the granting of high-level positions such as governerships of provinces, whom were directly answerable to the emperor and expected to constantly correspond with the emperor on their activities.

Associated with this role as "head bureaucrat" was the authority to deal with issues that cropped up in day to day running. Crime and punishment was usually the affairs of local officials, but certain crimes, such as treason and sorcery were typically immediately reported to the emperor. Just as the emperors kept tabs on the activities of the provincial officials, they could and did often take a close interest in particular criminal cases, closely corresponding with involved officials, dispatching other officials to mediate or even bringing the case to the capital for questioning. Furthermore, use of capital punishment had to be authorised by the emperor's direct approval and it was not unknown for criminals to simply die in prison of starvation, disease or wounds inflicted during torture while the official awaited imperial approval for the execution.

Beyond these roles, the emperor had final executive power in creating and implementing imperial policy. Law codes and other legislative sources could be modified by imperial decree, which could have enormous consequences for the empire. Famously, the emperor Dorgan promulgated a decree to the board of Rites requiring all male subjects to shave the front of their heads and wear their hair in the Manchu queue and restricting the wearing of hanfu, the traditional Han clothing. Despite resistance in many local quarters, the decree would prevail, and Chinese subjects would abide by these strictures until the end of the Qing dynasty (where the hairstyle would play an important role to revolutionaries as a symbol of an outdated and foriegn rule.)

So, far from being uninvolved and decentralised, the emperor played a crucial role in the state's management and running during the Qing dynasty. There is a lot I didn't cover here, particularly regarding the emperor's ritual and moral roles, which were just as central to the imperial institution as the more concrete roles I've discussed. Hope that gives you a bit more insight.

Is there any time period you're interested in? Given the thousands of years that this question spans, the role of the Emperor and bureaucracies will look very different if you're asking about the late Qing compared to say, the Song dynasty.