I‘m having an argument with somebody about this. I feel that the fight over the arguably legitimate line of succession in that show feels odd because it is lacking any meaningfully presented backdrop.
Its mostly about if first born girl or the later born boy is the legitimate heir, priority of gender or priority of firstness, both lines of argumentation are presented as equally valid. The debate is shown taking place in what feels like a vacuume, and as mere rhetorical strategies, and not as having much to do with broad stroke societal views about the proper gender for a monarch.
I think the show is lacking something in the presentaion of that question, but my friend says its pretty close to how those arguments would have felt around 1100 ad in europe.
Is there a sensible answer on who of us is more right?
The first thing that needs to be addressed here is that the debate in the show is not centered around a conflict of birth order vs. gender. It is a conflict between tradition, and Royal prerogative. As presented and referenced in the show (and in the song of ice and fire books) the tradition for the royal succession in Westeros is that the first born son is the king’s heir, period. The show is clear that up until this point the heir has always been the kings son, with the important exception of the previous succession, in which the king had no legitimate issue and the succession was decided by a vote of the nobles.
The central political conflict arises not because rhaenyra is the king’s eldest child, but because she has been specifically named (and apparently regularly confirmed) as the kings chosen heir and legitimized as such by the homage of the realm’s leading nobles. This is in direct conflict with the traditional succession practice that dictates the kings eldest son is the heir presumptive. This was all well and good when rhaenyra was an only child, but once her younger brother was born a central debate emerges. In simple terms the question is “to what extent is the king’s will able to override a century of royal tradition?” Can the king upend an established precedent, or is the succession an immutable law that can not be altered?
Now, the question becomes “how does this compare to real history?”
Martin has gone on record more than once that “a Song of Ice and Fire” draws inspiration from the Wars of the Roses. While I have not seen anything saying as much, it seems to me that the events of House of Dragons are likely based on an earlier English civil war, called The Anarchy which lasted from 1138-1153. In very broad strokes the conflict revolved around a very similar question of succession, although with some notable differences.
In brief, King Henry I was king of England. His only male heir, William Atheling, had died in the Wreck of the White Ship in 1120. He named his daughter Matilda, his only legitimate surviving child, as his heir and had the barons of England pay homage to her and swear to uphold her claim as his legitimate heir. King Henry dies (1135) and his nephew, Stephen of Blois, claims the crown with the support of a great number of the English barons. This kicks off a brutal, protracted civil war that is eventually ended when Matilda’s son, Henry II, is confirmed as co-king (a common practice in the period, Henry would later name is own son as his co-king) and sole heir of Stephen. When Stephen dies Henry II becomes king, an event that is usually considered the beginning of the Plantagenet dynasty in England.
Some quick differences worth noting. Stephen was not Henry’s son, although he had a reasonable claim to the throne as his male relative. Matilda was married to Count Geoffrey of Anjou at the time she was named heir and thus had a male consort to presumably fulfill the traditionally male-exclusive functions of kingship. Stephen was also in England at the time of Henry’s death and moved quickly to secure various aspects of royal authority, including the treasury, while Matilda was on the continent. Additionally, it was years from the death of Henry I until the beginning of the Civil war proper.
It is also worth noting that at the time of Henry I’s death England did not have a well established dynastic tradition. Henry I was only the third monarch since the conquest and the dynasty had not managed to create serious dynastic legitimacy the way the Targaryens seem to have done (probably because they didn’t have the benefit of fire breathing super weapons). Almost every royal succession up to Edward I a century later was heavily contested in one way or another. There was no clear tradition o succession, especially as Henry I had inherited the throne from his brother, William Rufus.
As mentioned, This strikingly different from the world of game of thrones where kingship seems to be pretty firmly established as an institution in and of itself. In the real world European ideas of kingship were nebulous and often poorly defined for a very long time, it wasn’t until the late Middle Ages that kingship became as institutionalized in Medieval Europe as it seems to be in Westeros. For a long time kingship remained a very personal business, especially in areas without strong dynastic traditions or legacies like post conquest England (or Norway as another example). Claims of blood and birth were important, but often times the man who became king was the one with the widest appeal on a personal level, or the one most widely regarded as capable of holding the throne. More often than not this meant the one who was the most successful militarily. There was a strong elective element to 12th century kingship and In the absence of a strong central government or administrative state the barons were the king makers, their support could make or break a claimants play for the throne. (Which is represented in the show by the oaths Rhaenyra is given by the realms leading nobles and by HighTowers scramble to secure their support for his grandson)
What this means for your question is that an early, or high medieval succession dispute would focus heavily on who was a more viable monarch. Formal, legalistic claims to the throne were often simply a way to legitimize a play for the crown, they were very much secondary to practical concerns like wealth, popularity and especially military power. So, in short, while House of Dragons has some elements of realism in its depictions of a medieval succession crisis it is perhaps a little to concerned with formal, legalistic claims to the throne rather than personal characteristics and qualifications for the throne. The show depicts personal relationships and the practical qualifications of the claimants as entirely secondary to their legalistic claims.
I hope that helps!
Edit: I realized I forgot to add dates.
There's more that can be said, but /u/swarthmoreburke wrote an excellent answer to How realist is the "apolitical struggle" in House of the Dragon? that addresses the issue from a historiographical perspective.