I would challenge this premise by saying that there were several attempts to 're-stitch' the empire and dozens that attempted to recreate a new Roman Empire.

First of all, in attempting to re-stitch you had players like Justinian who successly fully reconquered Italy (including Rome) in the 6th Century. He basically managed to reconquer the whole empire barring most of Gaul, Iberia, and Britannia. Following that, you continue along the Byzantine lines (who always claimed to be Rome) until the final dissolution in 1453.

Aside from that though, after Justinian, you additionally had more what we would 'recreations' of the empire. Charlemagne was the next most successful person in this list in the 8th Century. It's a little unclear as to where his exact heritage lies, however it was claimed he was descended from a Roman Governor of Gaul (many people claimed this though to add legitimacy to their rule).. Spanning from Francia (mostly modern day France and Germany), he also managed to capture most of Italy, but importantly he got Rome.

In an effort to legitimise himself further, he had the Pope anoint him Emperor of the Holy Roman Empire in 800, and that state in itself has its own vast and complicated history. Charlemagne also adopted the formula renovatio imperii Romanorum which means "renewal of the Roman Empire". Each subsequent Holy Roman Emperor from this point also claimed to be the true successors to the Romans although they had varying degrees of success. This was until 1806 when it was dissolved and succeeded by Napoleon and his Empire which, again, conquered Rome.

Other claimants of the successorship of Rome include Mehmed II, who, after the fall of Constantinople in 1453, declared himself Kayser-i Rum, or 'Caesar of Rome'. This claim rested with the idea that after the fall of the Western Roman Empire Constantinople, which he now ruled, had become the seat of the Roman Empire. At the time, his Kingdom was known as the Sultanate of Rum, which later became known as the Ottoman Empire which continued claiming to be the successor of Rome until 1922 when it was dissolved.

Mussolini is possibly the most modern despot to attempt to restore the Roman Empire. He of course ruled over all of Italy and secured some land from the Balkans and Northern Africa. Most the gains were ripped away in 1943 during the course of WW2 and he was deposed.

This isn't even an exhaustive list. There have been countless attempts at either restoring the Empire, or at least claiming to succeed the Empire. Other claims include those from Bulgaria, Greece, Russia, Serbia, Austria, and of course the Papacy.

(1/2) A bit of a difficult question to answer, especially because it is inherently counterfactual (what would have had to be different for the following to happen?) and spans a rather long period of time (the last millennium and a half since the end of Western Rome). One way to still find a solution might be to question the premise of the problem. Did the 5th century AD really mark the definitive end of imperial unification at the western end of the Eurasian landmass? To that I would answer: no, it didn’t. Both ends of Eurasia developed along relatively similar trajectories throughout antiquity and this didn’t really change all that much with the onset of the Middle Ages. First, however, we must abandon the idea that an empire that stretches from Europe across the Middle East and North Africa, like Rome, represents the normal state of affairs for a world empire at the western end of Eurasia.

When it came to empire-building Western Asia had a bit of a head start over the east of the continent. The region between the rivers Nile, Euphrates and Tigris, often called the ‘Fertile Crescent’, witnessed the emergence of agriculture as early as the 10th millennium BC, making larger, more complex human societies possible and allowing densely populated cities and more extensive state structures to develop over time. At the turn of the third millennium BC, the lower Nile valley was united for the first time by the pharaohs of Egypt. In the second half of the third millennium the Akkadians managed something similar along the banks of the Euphrates and Tigris, to be followed by a succession of Mesopotamian empires like the Third Dynasty of Ur or the Old Babylonian Kingdom. And in the 7th century BC, the Neo-Assyrian Empire was able to unite the entire Fertile Crescent into a single political entity for the first time with its conquest of Egypt.

East Asia lagged a little behind developments in the West. Still, between 8000 and 6000 BC it too saw the development of agricultural societies, this time along the Yellow River and the Yangtze. Here, too, the result was the emergence of cities and larger political units, in a period that traditional Chinese historiography assigns to the Xia, Shang and Zhou dynasties. A large number of principalities fought for control over the Yellow River and Yangtze region during the Spring and Autumn Period (8th to 5th centuries BC) and the aptly named Warring States Period (5th to 3rd centuries BC). This came to an end in 221 BC, when the state of Qin had overcome all its rivals and for the first time united the entirety of East Asia’s early agricultural zone. The Qin-Dynasty and its successor, the Han, didn’t stop there though. The armies of this first Chinese empire would march as far as Korea, Northern Vietnam and even into the far west of Central Asia, creating a polity that dwarfed any that had come before in East Asia by a large margin. With a few short interruptions this impressive achievement would last until the end of the second century AD. The unification of China came even slightly earlier than Rome’s conquest of the entirety of the Mediterranean. However, one should not conclude from this that East Asia had meanwhile overtaken developments in the West, for Rome was by no means the first empire of this size to develop at the other end of Eurasia.

The Achaemenid Persians had already conquered the entire Fertile Crescent as well as Iran, Anatolia and small parts of the Balkans and India by the end of the 6th century BC. A complex administrative system kept these expansive territories united until the death of Alexander the Great and the following wars of his Diadochi in the last decades of the 4th century BC. When China was united for the first time, the Persian Empire had already been gone for a century. By the end of the first century BC Rome had picked up many of its parts, combining the erstwhile Persian eastern Mediterranean with the rest of its Mare Nostrum and large parts of Europe. But Rome was not the only heir of the Achaemenids, for the Iranian highlands and most of Mesopotamia instead formed the core of the rival Parthian Kingdom.

The 3rd century AD was one of imperial crisis at both ends of the Eurasian landmass. Rome was shook by a large number of usurpations and for a short while even fragmented into three separate entities. In Persia the ruling dynasty of the Parthian Empire, the Arsacids, was toppled and replaced by the Sasanians. Meanwhile, the Han dynasty of China came to an end as well and gave way to three rivalling kingdoms, quite like the situation in the Roman Empire. All three empires were eventually able to overcome these crises, albeit with widely differing speed and long-term success. Persia bounced back almost immediately under the new Sasanian leadership and even turned towards a more expansionary foreign policy, thereby contributing to the problems of its Roman neighbour. Nevertheless, Rome was able to reunite relatively fast and the reforms of Diocletian and Constantine the Great at the turn of the 4th century AD put imperial rule on a much more secure footing here as well. By that time the Chinese Jin dynasty had equally been able to reunify the three post-Han kingdoms.

The 4th century AD turned out quite differently in East and West. Rome and Persia seemed to have found new strength and were able to turn much of this energy towards their mutual rivalry. In China on the other hand, the Jin dynasty already lost control of its northern core at the beginning of the century. What followed was a long phase of political disunity along the Yellow River and Yangtze, exacerbated by the involvement of nomadic peoples from the northern steppes, who were able to sack the Chinese capital of Luoyang in 311 AD and would later facilitate the foundation of several successor-kingdoms in the North of China. As the century progressed, Rome and Persia would feel increasing pressure from the Eurasian steppes as well. Different Hunnic groups posed significant military challenges for both empires, forcing them to put their own rivalry on the backburner. The crisis which then beset the Roman Empire with the onset of the 5th century had much more severe consequences than that of the 3rd century. Almost exactly a century after the fall of Luoyang, Rome was sacked as well (by the Visigoths) and not for the last time during the 5th century. In the end control of the entire western half of the empire had to be ceded to a variety of smaller successor-kingdoms – a development usually called ‘the fall of the Western Roman Empire’. In the meantime, Persia had to face the threat of the newly emerged empire of the White Huns / Hephthalites in Central Asia, to which it lost its eastern provinces, and which forced the Sasanians into a humiliating tributary position.

The 6th witnessed large-scale attempts at imperial reunification in Rome, Persia, and China. Under emperor Justinian the Eastern Roman Empire successfully retook large parts of the western Mediterranean, turning it into Rome’s Mare Nostrum once more. His great Persian contemporary, Khosrow I, could celebrate comparable triumphs by destroying the Hephthalite state to his north-east. And towards the end of the century emperor Wendi of the Sui dynasty was able to put an end to more than one and a half centuries of partition when he reunited China once more. In spite of these similarities though, the long-term success of the three attempts at imperial revival would vary significantly.