To some extent, the fictional frame of House of the Dragon reflects the degree to which a great deal of fantasy writing rests on an older approach to political history that is now uncommon among scholarly historians, namely, the kind of political history that concerns itself with conflicts between elites within formal political institutions that does not even address the degree to which those conflicts are produced by social structure (e.g., by discussing the class identity of the combatants). To a significant extent, this is what is commonly and dismissively referred to as the "Great Man Theory of History", which doesn't necessarily mean that all the people involved are laudable or remarkable, more that formal politics is a domain whose history can be reviewed as if the wider social context matters only as a policy problem facing leaders and not as an important causal driver of political events and conflicts in its own right.

The meta-issue in this case is that House of the Dragon is an adaptation of Martin's Fire and Blood, which presents itself to some extent as a historical chronicle created by the maesters of Martin's world--and thus resembling some of the kinds of chronicles written about many premodern dynastic histories by courtiers, monks, or other scholars. Many of these chronicles, such as Rashid al-Din's famous "world histories" that were written in part to exalt the rise of the Mongol Empire, tend to treat the formal politics of dynastic courts and the conflicts between empires or kingdoms as exclusively involving the activities of ruling families and their elite allies, with perhaps a bit of commentary on the side on broader social structures or practices. So if House of the Dragon seems not so much "apolitical" as asocial, that's a pretty good bit of mimicry on Martin's part--and markedly different from his main Song of Ice and Fire series, which shows quite a lot of attention to social structure and social hierarchy with a fair number of "ordinary people" showing up as characters as well as acting collectively through social, religious and political movements in ways that determine some of the political outcomes in the series.

Even in the case of Martin's series, however, the social structures of Westeros have been incredibly static over staggeringly long periods of time in comparison to real human history, which I think partly shows the reliance of fantasy as a genre on a very old-fashioned middle-school conceptualization of medieval Europe as an unchanging and static feudal society from the fall of Rome to the beginnings of the Renaissance. Very few works of high fantasy are built around highly dynamic social histories where the main dramatic conflicts are a product of that history (e.g., if there is social change happening, it tends to be happening for the first time during the story that's being told; it's not a backdrop assumption built into the setting).

In that sense, there's a very wide divergence between scholarly history and the kind of historical backdrop that informs high fantasy, because most scholarly historians who write about formal politics situate formal politics within wider socioeconomic contexts, or relate them to shifting cultural and intellectual histories. In that sense, I think there are very few practicing historians in the academy who would today treat any political conflicts in any historical era as "apolitical" in the sense that the OP means it--as motivated substantially by entirely personal feelings or entirely bounded by and contained within formal political systems themselves.

That doesn't mean that practicing historians exclude the personal feelings and character of important elite actors within formal politics, mind you. You can't write scholarly work about mid-19th Century British politics and ignore the personal antagonisms between Disraeli and Gladstone, for example, whatever else "British politics" was being caused by that had wider socioeconomic sources. A historian could choose to just say "for the sake of the narrative coherence in the following article/book, I'm going to foreground the personal relationships between political actors", and quite a few do just that (especially those writing for wider publics)--say, Doris Kearns Goodwin's Team of Rivals, about Lincoln's cabinet. But even there, Goodwin wouldn't argue that the personal relationships between Lincoln and his cabinet were exclusively determinative of wider political conflicts shaping the Civil War, that the Civil War was in this sense just an outgrowth of personal antagonisms between national-level politicians. I honestly cannot think of any history of past political conflict that a scholarly historian would today claim to be exclusively or predominantly a result of interpersonal rivalry or dislike that had no wider social, economic or cultural causality.

Since it's on my mind because of my teaching schedule this fall, for example, while I would certainly agree that dynastic rivalries and personalities were important causes of political conflict in the 14th Century Empire of Mali along the Middle Niger River in West Africa, I would argue that the disagreements that shaped dynastic struggle had much wider social implications--for example, the question of whether inheritance should take place through matrilineal or patrilineal descent, the question of how close the potential claimants to power were to Islamic trading networks and scholars, the legitimacy of the claimants within traditional hunting brotherhoods and among cavalry-based nobility, the popular support for claimants within Mande-speaking farming and fishing communities, kinship ties between dynastic rivals and some of their tributary courts, etc. I can't imagine just talking about all of those struggles in terms of the personalities and personal relationships of the dynastic rivals.

Arguably, if you were a historically literate person within Martin's world in the time of the events of the HBO series, you could in fact challenge the maesters who explained this conflict in terms of the rivalries of two branches of a dynasty. For the sake of drama, the show plays up Otto Hightower's coup attempt as a power-play that involves compelling the representatives of Westerosi noble houses to bend the knee in a closed chamber while also relying on the support of a small number of active conspirators. But a scholarly historian thinking outside the personalities involved (and the interdependence between maesters and the nobility) might wonder if Hightower's moves would hold up even if he moved fast and crushed the rival Targaryens--after all, at some point the new king would have to hold power through those nobles that he coerced. His Hand can't keep them in that room forever, and if there are broader, deeper dissatisfactions with King Aegon's rule (or Otto Hightower and his co-conspirators) then they might face a rebellion even if they managed to murder all their dynastic rivals. To some extent this is what Varys is talking about in the earlier HBO Game of Thrones series when he talks about power as a shadow, as something that commands obedience and respect only as long as people regard it as legitimate. In fact, a scholarly historian (even Martin's inside-the-world maesters) could see the events of House of the Dragon as paving the way for a long crisis of Targaryen legitimacy that ends up manifesting literally in the diminishing size of their dragons but which also is reflected in the attitude of Westerosi nobility and their dependent communities towards the Iron Throne. The show itself is not much interested (so far) in those wider social implications, just like the dynastic chronicles it is imitating/drawing upon, but any contemporary scholarly historian could imagine those wider issues just as much as we do when we read real-world political narratives that are seemingly "apolitical" in the OP's sense.