Firstly there are some issues with your question that needs to be addressed. The so-called “barbarians” that migrated into the Roman Empire throughout late antiquity did not have any kind of national identity. Most of them didn’t even have a shared cultural one despite coming from similar cultures and speaking related languages. People often have that view because the Romans tended to lump these peoples together into larger groups. So the Goths, Vandals, Marcomanni, and the Quadi might have been grouped together as “Germanic” people by the Romans but this is misleading because none of those groups would have referred to themselves as such. In fact some tribal groups, such as the Franks, were actually large groups composed of various tribes that the Romans just lumped together under one generic name. For example, the Tervingi Goths and the Greuthungi Goths were related to one another and both spoke Gothic (albeit different dialects) they were two separate groups of people that the Romans usually just referred to simply as Goths. This is a less egregious example than say lumping together all of the groups I listed previously, but it still shows how little effort the Romans put into understanding the diverse groups of people who lived across the Rhine and Danube Rivers. So while some of these groups may have occasionally raided the Empire or engaged in outright warfare with the Romans for some kind of “revenge”, it only would have been done after the Romans attacked them personally.

The next problem with your question is your usage of the term “invasion”, the great migrations of late antiquity were absolutely not invasions. These were migrations of people who wanted to move into the Roman Empire so they could live under its protection, not because they wanted to cause its demise. It is true that many of these migrations led to conflicts and battles because the Romans of course weren’t in the business of just letting tens of thousands of people settle inside their borders anytime they felt like it, but the groups who migrated during this period were eventually allowed to settle in areas that had been depopulated. These peoples would in turn send men to serve in the Roman military, especially the Goths. (I apologize if this sounded pretentious I just think its important to correct things like this when possible, because it is a rather common misconception.)

That being said, there is actually one example of an outright invasion during this period. That being Radagaisus’ invasion of Italy in late 405. Radagaisus was a Gothic king who led a group that may have been as large as 100,000 people (most of whom would have been noncombatants). But even this group was fleeing the Huns and so they only invaded Italy as a means to escape being conquered themselves. As far as I’m aware we don’t have any real idea as to what Radagaisus’ end goal was, but it likely couldn’t have been very different than Alaric’s goals a few years later when he invaded Italy. That is, he probably wanted some kind of assurance from the emperor that his people could settle land somewhere inside the Empire, likely in exchange for sending men to serve in the military, and perhaps even some kind of official position in the military for Radagaisus himself (plenty of non-Romans already held high ranks in the military, such as Stilicho who was half-Vandal).

Now the part of your question that is true is that the Romans did often depict these migrating people as “villains”, to use your own term, when writing about them. When discussing the Goth’s (in this case the Tervingi Goths) crossing of the Danube with the Roman’s help, Ammianus Marcellinus wrote “And the greatest care was taken to ensure that, even if they were suffering from a mortal illness, none of those destined to overthrow the Roman Empire should be left behind.” (Ammianus Marcellinus, Res Gestae 31.4.5) Clearly Ammianus, like many Romans, held biased opinions of non-Romans such as the Goths. Granted Ammianus did write his history after the Battle of Adrianople, where the Goths inflicted staggering casualties on the eastern field army, the Goths played no role in “overthrowing” the Empire and were settled within its borders after Theodosius I defeated them. Ammianus was also writing during the reign of Theodosius I and may have been playing up the threat of the Goths to make Theodosius look better since he’s the one who defeated them in battle.

Ammianus also described the Huns, a people he almost certainly never had any contact with, in rather derogatory ways:

They are certainly in the shape of men, however uncouth, but are so hardy that they neither require fire nor well-flavoured food, but live on the roots of such herbs as they get in the fields, or on the half-raw flesh of any animal, which they merely warm rapidly by placing it between their own thighs and the backs of their horses. They never shelter themselves under roofed houses, but avoid them as people ordinarily avoid sepulchres as things not fitted for common use. Nor is there even to be found among them a cabin thatched with reed; but they wander about, roaming over the mountains and the woods and accustom themselves to bear frost and hunger and thirst from their very cradles. And even when abroad they never enter a house unless under the compulsion of some extreme necessity; nor, indeed, do they think people under roofs as safe as others. (Ammainus Marcellinus, Res Gestae 31.2.3)

Now obviously the Huns did not subside on roots or uncooked meat and would have used fire for various purposes. They also had proper homes, contrary to Ammianus’ beliefs, but they were a seminomadic people so they might have only lived in these homes during the winter when their herds weren’t grazing in summer pastures. There is also some archeological evidence that they grew some of their food as well. So they were far from the uncivilized nearly subhuman beings that Ammianus described them as.