Seems confusing right? All of a sudden there were these Ostrogoths running Italy and Visigoths running Spain, but they seem to evaporate in a cloud of smoke over less than a century.

Although I'm pretty sure you can trace a line straight from the Visigothic Kings to the Castilian, Aragonese, and Portuguese monarchies, I actually don't know all that much about Iberia. What can speak to is the political and ethnic identity of the early Italian ruling class in the late fifth and early sixth century.

I think John Moorehead's 1992 "Theodoric in Italy" gives you the best idea of what the Ostrogoth elite was like right when they were most influential, but for a more holistic overview you can look at the more recent "The Ostrogoths from the Migration Period to the Sixth Century," a 2007 collection of papers edited by Sam Barnish and Federico Marazzi. Jonathan J. Arnold's 2014 book "Theoderic and the Roman Imperial Restoration" also offers a more recent, and fresh, look at the history and histerography of Italy in the very late antiquity.

So I'd start by pointing out the nuances of the relationships between "Barbarians" like the Ostrogoths and the Roman Empire. Older historians, and modern "Common Knowledge," narrate a unified Roman Empire locked in violent confrontation with barbaric invaders set on dismantling their civilization. This narrative dates as far back as the Italian Renaissance, although some would argue say it has its embryo in the rhetoric of Emperor Justinian's reconquest (more on that later). But the narrative in modern academia (and, as a consequence, here on AskHistorians) is much different: we discard the idea that there is a definitive break between the ancient world and "Late Antiquity." Consider how modern scholars treat the late Roman frontier as a broad microcosm: as the Roman borders became static, the legions became increasingly reliant on local recruitment. This, in turn, set the stage for local power structures to develop, and as these structures came into conflict this turn sets the stage for the middle ages. But although there certainly was a degree of crisis and disruption, the settlement of most "barbarians" in places like Gaul was in fact rather orderly; and there is a fair degree of continuity and collaboration between groups of people like the Franco-Gaulish military elite and the Gaulo-Latin administrative class.

Amidst the clear loss of Imperial authority in Western Europe, Italy had a unique place. Although it was the central and most urbanized part of the Empire, legions hadn't been actually recruited there in earnest for centuries. Rome, although infused with symbolic significance, yielded its practical prominence to other cities like Ravenna and Milan. The increased autonomy, and eventual breakaway, of the provinces had left a dark could over the entire spectrum of the Italian elite. From north Italians like Magnus Felix Ennodius, to southerners like Flavius Magnus Aurelius Cassiodorus, the general mood in the writing of the time was bleak (but was it as bad as they described? Do we trust them? Both individuals were subsequently privileged by Theodoric's upheaval). What is unequivocally clear is that for these individuals, the ascension of Flavius Odoacer to the position of Rex Italia, or King of Italy, did not signal an end of the empire or an end to Roman-ness. Odoacer was a bad ruler, but not because he was not a Roman (to this day, we're not even sure of his ethnicity; and consider that "Flavius" is a very Roman-sounding first name). Rather, Odoacer was only the latest bad ruler in a succession of bad rulers. In fact, some arguments were made at the time that rulers and authority figures with "barbaric" origin were more "Roman" than the "Greeks" sent to help by Constantinople!

So here enters Theodoric, Consul in Constantinople and Leader of the Ostrogoths. Yes, those two jobs were not mutually exclusive. By the time Theodoric was appointed Consul, the Goths had been living Pannonia, a region along the aforementioned broad border zones of the Empire, for centuries. In short, they were thoroughly Romanized. We are almost positive Theodoric himself was a speaker of Greek as a first language. There doesn't even seem to have been much solidarity and unity within the "Gothic" blocks in the Eastern Roman army, seeing as they fought against each other, and sometimes against the Emperor, in a multitude of combinations.

The Eastern Emperor Zeno, who himself ruled on a flimsy pretext, grew increasingly uneasy as he watched Odoacer usurp his counterpart in the west. When Theodoric restored order within the Gothic ranks on the cusp of a civil war, Zeno couldn't have helped wondering if his own removal was also forthcoming. And if Theodoric or some other general were to oppose Zeno, where would Odoacer's loyalties lie?

Zeno resolved to take out two birds with one stone: Send Theodoric and his men to conquer Italy and depose Odoacer. A plot to reaffirm Roman control over Italy? Not really; more of a plot to place someone more desirable at an important post that was nonetheless far from Constantinople.

More than a horde of barbarians, some assessments of Theodoric's following go as far as to describe it as exclusively composed armed and unaccompanied young men, some of whom were not even Goths. In short, what is described looks suspiciously like a professional army (Patrick Amory is a notable proponent of this idea, although his writing is on the most extreme end arguing for seamless post-Imperial societal continuity; other scholars cite evidence of more women and children in tow).

Even before his conquest and usurpation was complete, the Italian aristocracy absolutely fawned over Theodoric: he's cultured, refined, intelligent, but also possesses a military virility in keeping with the "Old Roman" values (and he also extensively taps the Latin upper classes for important positions of power). The continuities and changes of Theodoric's reign can be summarized, as Arnold does, as "overwhelmingly Roman and Imperial in nature." Indeed, the entire third part of Arnold's book examines the notion of "Gothic-ness" and makes the, nowadays predictable, conclusion that the Goths were in reality defenders of the Roman Empire and the central pillars of "Roman-ness." In fact, "Goth" seems to mutate into a military-societal designation, with Theodoric himself commenting (perhaps apocryphally) "A poor Roman plays the Goth, a rich Goth the Roman."

So there you have it. The ethnical successors to the Ostrogoths, such as we can even say they existed, would be the dispersed among the society that subsequently emerged in Italy.

I wrote a longer, more narrative answer here on a similar topic which you might be interested in. I also wrote a shorter answer on the integration of the Lombards in Italy; like the Ostrogoths, the Lombards were similarly integrated very rapidly into Italian society, and the word "Lombard" even became shorthand for "Indigenously Italian." I also wrote an answer on the broader issue of Italian identity here.

Lastly, a more historiographic discussion to a very similar question coming to very similar conclusions is here.