While I’m not an expert on Germany or German Things, I think I can still lay down some ideas linked to the early history of the Holy Roman Empire which itself is very much linked to the History of Italy (an area which I do know a thing or two about). The short answer to your question core question of, “When did Central European powers become positive or envious of Rome rather then seeing them as a foreign conquering power?” is, “Sometime after the Crisis of the Third Century and long before the end of the Roman Empire itself.” To offer more depth, we can answer this question in two ways: First, To what extent were Germanic peoples living along the border Romanized? And second, why is is that once autonomous, these peoples (however Romanized they were) continued using some semblance of Roman titles? I’m going to take the second approach: I’ve written a short history of the use of the title of “Emperor” in Early Medieval Europe below.

To start, it might be useful to clarify that when the “Holy Roman Empire” came about, it emerged in a world which had seen nearly a millennium pass since the Ancient Romans marched (with spotty success) to subdue the peoples of “Germania.” And it’s a now-common trope to emphasize that although the final centuries of the Roman Empire were indeed rife with conflict and instability, this instability was not particularly damaging to institutional continuity: we are discussing a world with centuries of interaction between “Roman” society on one side of the border and “Germanic” society over the other side of the border; and when a Frankish Kingdom emerged straddling both sides of Rome’s old “Germanic Border,” the old Roman institutions were actually some of the fundamental legitimizing tools that the Franks exploited. This, in short, the reason why the Franks (and later the Germans, however reductive that term is) were comfortable with “Roman” pageantry to the extent that they were.

I also think it would useful to start with the pretext that notions of “Empire” in the sense that the Imperial-era Ancient Romans intended doesn’t really fit into out modern understanding of sovereignty and statehood, and this notion persisted into the medieval era. Although the Roman’s own abstract understanding of their Empire did change over the centuries, they generally understood it as something universal: The empire in effect had no borders, only places where the Roman way of life had yet to be adopted. Frontiers did exist, but that doesn’t mean the Romans recognized anyone else’s sovereignty: frontiers had to be guarded in order to defend people living the Roman way of life against people who were not (yet) living the Roman way of life. The Emperor’s authority was likewise universal: if Romans might have actually wanted to rule the whole world is irrelevant, they revealed themselves indifferent to allowing client rulers to exist insofar as those client rulers subjected themselves to the Emperor. Where the Empire encountered rulers that did not respect the Emperor’s universal authority, the Romans marched to war and instilled a ruler who did (sometimes they were succesful, sometimes they were not; these were practical limitations, but did not change the underlying assumption that the Emperor had the right to rule the world). This notion is important because it explains, in part, what the title of “Emperor” implied even after the Empire disappeared.

This line of thinking was largely kept intact after the fall of the Western Roman Empire: the first Kingdom of Italy explicitly acknowledged the primacy of the Emperor in Constantinople (affirming imperial universality) while two centuries later the second Kingdom of Italy was locked in an existential struggle with the Emperor in Constantinople (also in a way affirming imperial universality, in that the only way to exist outside the Empire was to be in conflict with it).

It is only in the eighth century, when the Eastern Empire proved unable to defend Rome (a city still imbued with symbolic importance, albeit in serious decline with regards to all other measures of importance) and the city itself was beset by turbulence and internal strife, that the city’s leadership (partially but not entirely consisting of what remained of the old senatorial class) turned and pointed to the Frankish King to offer a sort of “Imperial” mandate (yes, the Papacy did lead the efforts to this end, although the Papacy wasn’t really something we would recognize just yet).

It had taken about two generations for the ruling class in Rome (and other parts of Italy as well, but mostly in Rome) to complete the gradual realization that their social order was no longer going to be protected by whatever they were calling “Empire.” Indeed, over the course of those two generations Frankish rulers had proven much more valuable allies in defending Rome’s primacy and autonomy against encroachment by Kingdom of Italy. We can, if we’d like, open up a short speculative parenthesis and ask ourselves what the second Kingdom of Italy would have actually looked life if King Desiderius' army (or whichever Lombard King) had succeeded in seizing Rome: lots of sacking, probably, but not a lot of wholesale change in terms of social organization (the second Kingdom of Italy’s ruling class were Lombards, originally a germanic people, but the size of that ruling class was so small and their adoption of Latin culture so earnest that they probably couldn’t change much in Italy even if they wanted to). We can, however, lay this parenthesis aside as irrelevant: cultural continuity or building a stable political organization on the Italian Peninsula was not something that interested the Romans: the ultimate objective for the Papacy and what was left of the old senatorial class was to preserve themselves through the final phases of a long, arduous, and immensely destructive conflict between the old “Empire” and a new Kingdom. In a way, seizure of Rome by a Lombard King would cut off what was presumably (but of course not practically) the seat of the Empire from the Empire itself, something that for Rome’s leadership was fundamentally tantamount to the end of the world. So the eventual solution was to flip a switch: the Emperor was not in Constantinople, the Emperor was seated somewhere else, and that someone else would have an easter time defending the city of Rome.

The choice of the Frankish king was not arbitrary. Sure, the monarchs themselves spoke a germanic dialect (Franconian) but the language of administration was the earliest form of Old French (or as our eighth-century Romans would have perceived it, Latin spoken by Franks with bad grammar; was this much different than Latin spoken by Greeks with better grammar?). The Frankish monarchy relied heavily on religious pageantry for their legitimization and a component of the Frankish clergy had always been in full communication with clergy in Italy. This was, in other words, not really a group of people who had very much at all to do with Ancient Rome’s enemies in “Germania” centuries ago. Indeed, setting aside the centuries of Frankish settlement inside the Empire’s borders (and Frankish recruitment in the Roman Army) in the several centuries since the fall of the Western Empire the Frankish monarchy’s success had in great part relied on the successful appropriation of the old roman institutions of both the frontier and the Gaulish interior, which coupled with their undoubtably successful military organization allowed them to expand the borders of their Kingdom eastward. Thus rather than an alien Germanic people, the Franks were instead a a (semi-)Latinized people who had in fact imposed Roman(-ish) rule in a region where actual Romans had never been able to successfully do so. “Good enough” resolved the Romans, not without a hint of begrudging admiration.

What "Empire" meant for of this new "Carolingian Empire" and its impact is discussed in the following comment.