Some of these events have been exaggerated or misrepresented, but there is a pattern that emerges when we consider them together:

1. The scale of the Gaulish sack of Rome was probably exaggerated by Roman writers who wanted to portray Rome as staunch enemies of Gauls for a wider Mediterranean audience. Archaeologists have found no evidence of substantial destruction at Rome that can be dated to the supposed Gaulish sack. If the event happened at all and is not entirely a fabrication of later writers, it was probably no more than a raid for plunder of the sort that Italian peoples of the time--including Romans--routinely made against one another.
2. Hannibal's overall strategy was to strip Rome of military power by detaching its Italian allies and subjects. A significant number of southern Italian cities joined Hannibal, but these cities had mostly been recently conquered by Rome and at the time contributed little to its military force. Rome's core allies in central and northern Italy, which made up most of its fighting force, remained loyal to Rome.
3. The civil wars and violent political clashes of the last century of the republic were primarily conflicts over who should control Roman politics. Few of the combatants had any interest in destroying the Roman state and its empire; they wanted to take it over and enjoy the privilege and prestige of ruling Rome, not tear it apart. The most significant threat to the unity of the empire in this century was the Social War, during which many of Rome's Italian subjects rebelled and at least contemplated creating an alternative state, but were ultimately satisfied with a promise of Roman citizenship.
4. The Roman Empire in the west disintegrated gradually over several centuries for many reasons. Civil wars and political turmoil in the third century broke down the political ties holding the provinces of the empire together and encouraged the growth of local political structures that were more effective at providing for defense and dealing with local problems. Outbreaks of disease drove people out of cities, which disrupted trade routes and loosened the economic connections between regions. Political and economic changes prompted farmers to focus on subsistence and self-sufficiency rather than growing cash crops, which further weakened trade ties. Imperial patronage became less reliable and less rewarding, which encouraged provincial elites to build up their local networks of power at the expense of loyalty to the emperors. Some individual emperors, like Aurellian, brought temporary stability and began to rebuild some of these weakened links, but the long-term trend in the Roman west from the third century on was towards disintegration of imperial control and strengthening of local political and economic structures.

One of the keys to Rome's success as an imperial state was its ability to incorporate other peoples, to build bonds of loyalty that kept people invested in the success of Rome. These bonds were too strong for Hannibal's strategy to work or for the chaos of the end of the republic to shatter. Loyalty to Rome was not simply sentimental; it brought tangible, material advantages that were worth having, even worth fighting for. When these advantages were eroded by the chaos of the third century, however, people were willing to let their bonds to Rome wither. The personal efforts of later emperors were not enough to reverse the trend toward fragmentation and local self-sufficiency.

(5 and 6 are beyond my area of expertise. Someone else will have to comment on those events.)

I would argue that to a large extent this is not chiefly a question of what happened but how events have been framed retrospectively to be as romantic as possible.

1. It is not uncommon to see a heroic narrative develop where the state is all but destroyed and then rescued from the extreme brink by one leader who represents all the martial values that it was felt its people should aspire to - but it is uncommon for this to be the clear-cut truth. As with almost all characters from early Roman history (including every king), we have no evidence for the full story of Marcus Furius Camillus outside historians from much later, like Livy and Plutarch, and it is difficult to tease apart truth from fact. We know that Rome had major conflict with Gauls, and we have other attestations of Brennus, but to what degree their account of the Battle of Allia is true is up for debate: certainly no level of sacking or destruction shows up in the archaeological record at that period, with no clustering of buildings having been destroyed then.

2. The story of the Second Punic War is certainly astonishing and had a great deal to do with establishing Rome’s power and later pride. I’d put this down as the most remarkable example here, but actual military analysis of how close Rome truly came to destruction I’ll leave to someone who has more in-depth knowledge of the archaeology, military strategy, and reliability of the traditional sources.

3. Most empires and states that have lasted any significant period of time have seen civil wars come and go. Ultimately, if a state is split into zones, whoever achieves power in the most powerful zone has a clear advantage, and reunification is always a powerful incentive. Octavian had his power base in Rome itself, which had both the political legitimacy of the organs of the a republic and the loyalty of the core of the military. This doesn’t fit the narrative of the previous two quite so easily.

4. Similar applies, though it was far more turbulent, to the Crisis of the Third Century: Rome still controlled the majority of the Empire, with the so-called Gallic and Palmyrene empires being relative fringes that never had the same level of consolidation, legitimacy, territory or military strength, and were even more unstable. But it is fair to note that the standards of salvation here would not be met by the fragmentation of the Empire that occurred later, with the fall of the Western Roman Empire (even then, never as clear-cut as much later historiography regarded it - the deposition of Romulus Augustulus not being an end to Rome as an idea, and the very idea that these would be equated being almost blasphemous centuries earlier), nor by the gradual eating away of the Eastern Roman Empire until it consisted of very little outside Constantinople itself.

5. The Empire did lose to the Persians - many times, even badly on occasion - as Valerian and Julian would attest. The war won by Heraclius wasn’t the determiner of the Empire’s continued existence, but fought chiefly in the Middle East. That said, both sides exhausting each other laid the groundwork for a far more destructive defeat to the Caliphate, which while destroying the Sassanids also lost the Romans the Levant and North Africa. This seems if anything at best cherry-picked and at worst the very opposite of a salvation.

6. Fair to note that here you’re talking about a completely different group identifying as ‘Romans’ from (1), whose domain doesn’t even include Rome (maybe we could have added the re-conquest of Italy by Justinian, Belisarius etc., as temporary as that was) - and in fact its enemies in this case, Venetian-led, are the ones called ‘Latins’ (!). This doesn’t make the strongest case for continuity. Even taking the Eastern Roman Empire as the political heir rather than an entity divorced from the city of Rome itself, this is framed as the Empire of Nicaea being The One Continuation of the Eastern Roman Empire while the Latin Empire was not part of that, when we could also see that as a new dynasty of the same - of all examples here, this is clearly a case of ‘history written by the victors’. Even in 1204, the Laskaris family were not the clear-cut heirs to the purple, but in 1261 had every interest in framing their conquest of Constantinople as a ‘restoration’. Chief among the reasons for this is the ethnic (Greek-speaking) and religious, rather than any political, continuity. But this is certainly a very inconsistent criterion when the original Romans were themselves very much Latins, not even Christian and… in Rome in Italy… and in between had been ruled by dynasties with several ethnic backgrounds. I don’t think that pagan, Latin-speaking adherents of the Republic at the time of Scipio Africanus would look at the Greek-speaking Christians of Constantinople under their emperor and agree with the assumed continuity simply on the basis that Constantine, a Christian emperor, declared Byzantium his new capital and his despotic successors split it along those lines. They would see that as a betrayal of every ethnocentric, religious and political value that made Rome Rome. It is only retrospectively that we see things this way.

Ultimately, early myths of brinksmanship and heroism, from Romulus to Horatius Brutus, Cincinnatus and Marcus Camillus, may have some basis in reality or may be myth, but are very likely at least exaggerated to fit this narrative. In between, Rome itself had a massive power base, and was able to defend itself. Later, Rome did not really maintain its power or survive in any recognisable form, especially not around the original city: but the idea of being heirs of Rome was extremely valuable to whoever wielded political power, especially in Constantinople. When the Holy Roman Emperors, Russians and even Ottomans and others also laid claim to this, it’s no surprise that states ‘closer to home’ (which, in a sense, included two of these in any case) would too. As for the Ottomans: it’s telling of our inconsistent criteria of what claims we accept that we consider ‘Rome’ to have continued past 1204 but not into the 20th century)

There also seem to be inconsistent standards where notions of defeat and ‘destruction’ would apply to plenty of other episodes in any case, which Rome did not escape: 410, 476 (if less materially and even debatably), the 640s, 751, 1071, 1204, 1453… the gradual reduction of Rome for a long period to a very small malaria-infested town… Absolute, permanent destruction of a major city - or broadly related idea, given we aren’t talking about the same thing across this period - is rare. There is even a town of ‘Sparti’ today. Perhaps the continued existence of so large, diverse and discontinuous an idea as China or the Caliphate are a better analogy here, but we haven’t obfuscated those with the survival of a plucky little city. Certainly the prestige of Ancient Rome in Western historiography and education would often like to emphasise their divine fortune, and maybe it would have been another city and name if not them - but compared to other massive empires or self-fulfilling concepts, how lucky were they, really?