A more accurate description would be that center of Roman civilization shifted from Italy to Eastern Mediterranean long before that (Wikipedia says that population of Rome fell from almost a million to mere 30 thousand in Late Antiquity, making it really just a minor town before the Barbarians moved in).
Roman civilization had had several major centers. The ones in the West gradually ceased to exist. That's the only sense in which the center of civilization "shifted". Some wealthy citizens of the city of Rome may have fled east, but the vast majority of the population of the western empire (Italy, Gaul, Iberia, Britain, Africa, not to mention the western Balkans and adjacent areas which were also conquered by barbarians in the 4th century) were agricultural and could flee only if they left all their posessions behind.
IOW, the fall of population by 60-80% in these areas during the 4th and 5th centuries wasn't accomplished by emigration. (Not to mention the immigration of barbarians.)
As for the city of Rome, it was sacked by barbarians in the years 410 and 455. WP suggests that its population declined from several hundred thousand to 80,000 during approximately the fifth century, but this is unsourced and I would like better information. At any rate, at the time of the 410 sack the population was already far below its 2nd century peak of 2 million. By the 4th century the emperors didn't live there anymore (some of the 5th century ones apparently did though), so its decline started before the invasions. Still, it was much more than a "minor town" in 410, containing many riches to plunder and rich and noble people to hold for ransom.
All in all, the Roman Empire did collapse. In ~400 the Western parts of the empire existed as it had for >200 years. By 450 it was effectively restricted to Italy and parts of southern Gaul, and in 476 it was officially terminated by the death of the last Western Emperor.
Compare this map of the entire empire in 117 (not much different than in 400). That's a loss, inside 60 years, of all of Europe west of the Balkans (including Italy), and all of Africa west of Egypt (the province of Africa, around Carthage, had been a major source of agricultural produce).
The Eastern empire did reconquer some of the West in mid 6th century. It lost half of that again by 600, and most of the other half by 650. In any case its rule there wasn't very much like the original Roman system in terms of culture (the barbarians were the local rulers) or economics, taxes and representation.
at least until battle of Manzikert in 1071 the central parts of Roman (Byzantine) Empire were doing just fine.
Those central parts were on the order of one-twentieth the area ruled by Romans pre-collapse, and many of them were to the east of the original Empire. Just because they preserved unbroken political succession and the name of Romans doesn't mean we should identify them with the original Empire.
In ~400 the Western parts of the empire existed as it had for >200 years. By 450 it was effectively restricted to Italy and parts of southern Gaul, and in 476 it was officially terminated by the death of the last Western Emperor.
Not quite accurate; in 376 a big bunch of barbarians half-forced, half-negotiated their way into the Empire, became disloyal subjects, and subsequently pillaged the Balkans and defeated killed an (Eastern) emperor and his army. So it's better to say that the Western Empire declined almost entirely during the 100 years 376-476. (Politically, militarily, and on a local rule level this is true. Culturally the collapse did take longer in some places.)
I really liked Robin's point that mainstream scientists are usually right, while contrarians are usually wrong. We don't need to get into details of the dispute - and usually we cannot really make an informed judgment without spending too much time anyway - just figuring out who's "mainstream" lets us know who's right with high probability. It's type of thinking related to reference class forecasting - find a reference class of similar situations with known outcomes, and we get a pretty decent probability distribution over possible outcomes.
Unfortunately deciding what's the proper reference class is not straightforward, and can be a point of contention. If you put climate change scientists in the reference class of "mainstream science", it gives great credence to their findings. People who doubt them can be freely disbelieved, and any arguments can be dismissed by low success rate of contrarianism against mainstream science.
But, if you put climate change scientists in reference class of "highly politicized science", then the chance of them being completely wrong becomes orders of magnitude higher. We have plenty of examples where such science was completely wrong and persisted in being wrong in spite of overwhelming evidence, as with race and IQ, nuclear winter, and pretty much everything in macroeconomics. Chances of mainstream being right, and contrarians being right are not too dissimilar in such cases.
Or, if the reference class is "science-y Doomsday predictors", then they're almost certainly completely wrong. See Paul Ehrlich (overpopulation), and Matt Simmons (peak oil) for some examples, both treated extremely seriously by mainstream media at time. So far in spite of countless cases of science predicting doom and gloom, not a single one of them turned out to be true, usually not just barely enough to be discounted by anthropic principle, but spectacularly so. Cornucopians were virtually always right.
It's also possible to use multiple reference classes - to view impact on climate according to "highly politicized science" reference class, and impact on human well-being according to "science-y Doomsday predictors" reference class, what's more or less how I think about it.
I'm sure if you thought hard enough, you could come up with other plausible reference classes, each leading to any conclusion you desire. I don't see how one of these reference class reasonings is obviously more valid than others, nor do I see any clear criteria for choosing the right reference class. It seems as subjective as Bayesian priors, except we know in advance we won't have evidence necessary for our views to converge.
The problem doesn't arise only if you agree to reference classes in advance, as you can reasonably do with the original application of forecasting costs of public projects. Does it kill reference class forecasting as a general technique, or is there a way to save it?