I don't think you need to worry or revise what you already have.I'm also not sure if the four founding dragons ever had any dragon offspring. They spent about two hundred years advising the nation of Swynfaredia, then they flew off. Over a thousand years later, the four founding dragons are practically worshiped They would be dead by old age by now, but hypothetically if a dragon showed up and claimed a direct blood line to one of these dragons flew in, he or she could start a civil war just by sneezing.
I could say that the four founding dragons were all dragon-celibate but I'm not sure if that's realistic and I'm sure whether or not it's good idea to include new draconic blood into Swynfaredia.
This is a feature, not a bug.
Cultural institutions are a patina of understanding built over time, and often contain assumptions that aren't sound as society extends from its original environment into a broad perspective. Dragons inventing a human(oid) society built to suit their opinions and establishing themselves as literal parent-kings, but not considering how their draconic line would interact with the human line....that's believable. That at some point in thousand-year history of the society, there was a sloppy patch-over to solve a problem, or the problem had to be resolved after a major disruption, or hasn't been resolved actually makes more sense than complete cultural stasis.
Especially since what you've described is a top-down social system imposed by fiat--the base premise of the hierarchy is arbitrary; there is no relationship between sorcery talent or dragon descent and the skills required to operate a nation, and the rest of the culture--who gets resources, who gets educated, who gets titled--bends to make that flawed base premise "true." It's four dynasties maintaining their power base...including through strategic marriage...except the qualifying condition is magic and descents from dragons rather than descent traceable to a legitimate heir with some other justification structure (descended from a god, from a blessed family, first king of the nation, first king of the "right" religion").
So technically it's not just the dragon's dragon offspring that could try and argue a claim, it's their parents, siblings, nieces/nephews.
Furthermore...once the base premise of "dragon descent is good" has been set, there's going to be a dragon out there who looks at the racial hierarchy of the country and sees two points of failure. The first is that there's nothing material stopping another dragon from joining the culture and starting their own dynasty' the second is that the needs for dragon-descended sorcerors is so incentivized by the racial hierarchy that regular re-infusions of draconic heritage is a very smart thing for the dynastic houses to reach for.
There's many, many holes in the base premise of the society...but that's not because you've written it wrong, it's because imaginary distinctions imposed by powerful people to preserve their power tend to be ridiculous and full of loopholes and technicalities, because the point isn't coherence, it's creating a rhetoric that serves the needs of the already-powerful.
It makes perfect sense to pay attention to those flaws and let players use those flaws to alter or destroy this society.
If you don't want this to be a factor in the setting, the simplest solution is: the society is racist, and could probably solve these cultural problems with more racism. Racists love fractally precise tiers of distinctions, and a natural distinction would be "relative dragons and half-dragons are somewhat special, they're allowed this much license within the society and their offspring can attain this rank; stranger dragons and half-dragons are permitted far less license and their offspring are allowed only to attain this lesser rank."
But...you seem to want things to be more dynamic!
Given a thousand-year history and the enormous incentives to have draconic heritage, it's far more likely century by century the ploy of bringing in new dragons and half-dragons has been tried both by the powerless and the powerful. Furthermore, a millennium is another enormous amount of time for a cultural or political entity to exist stably, so it would make perfect sense for there to be points of social upheaval or disruption that attempt to alter the power structure.
For example, the death of each of those founder-dragons creates a power vacuum inside their clan, between the four ruling clans, and between the ruling clans and the subordinate and subjugated peoples.
It would be perfect time for a dragon to stake a claim through kinship to be the next dragon-leader of the clan...this society is based on a racist tier system, and within that racist tier system it could be argued that a dragon kin to the founder is better than a humanoid kin to the founder...and this doesn't have to be a legal argument, much in the way that a bastard that clearly resembles their noble parent can barge into the succession when the law points towards some distant relative that technically inherits.
And if you were a clan leader in a society where there's always been four dragons in charge and that's the base assumption of where authority comes from...when your dragon in charge dies there's going to be a very rational fear about losing power, because the parity that shapes the society has been punctured, and there's no actual guarantee that the agreed-upon system will operate according to it's premise. So attempting to re-create the framework of the previous system but with a different dragon is...a reasonable thing to try within the unreasonable framework of the society.
It would also be moment where non-clan entity could attempt to assert themselves by allying with a dragon and re-creating the dynamic that established the nation...again, arguing the continuity of form is what is most important.
The intricate norms and laws that were discussed the last time you posted about this setting would likely not be the product of one giant bout of legislating, but generations after generation of trying to use culture and law to seal up loopholes in the laws about racial hierarchy and "creative" interpretations of societal norms about dragons.
Another likely point of crisis in this society is if there's a generation where no one at the top of the social hierarchy meets the culturally established criteria for right-to-rule...which is entirely feasible over 33-someodd generations, especially if these people are in-breeding to maintain "bloodlines." And it's not just biology--oligarchies create intellectual and cultural cloisters in which the social intrigues of the powerful are more important than the meat-and-potatoes practical, so after generations of infighting the top echelon has to be skilled at petty infighting to get power, but doesn't necessarily have any skills to hold power. A war, plague, or accident could drop an enormous number of legitimate candidates, too. So if the culture criterion of rulerships creates a situation of rule by scrubs, incompetents, weirdos, the basic premise of the society is exposed as...dumb.
Individuals more practically qualified to rule--that have economic, political, and military skills--will start questioning the very idea of the hierarchy. Individuals with some of the cultural qualifications the society has elevated--skilled sorceror, more dragon--not of the traditional family power structure will also question the premise of the society.
The society you've described has fetishized dragons. Dragon-ness is validity and thus the gateway to power, and since the powerful are supposedly more dragon, dragon-ness (haha, dracosimilitude) will also be...beautiful, elegant, a sign of refinement, reflective of a "superior" mode of being. So while the law might try to hammer down that only the ancestral dragon line matters, and the top-down culture emphasizes the same...the presets of the culture mean people want to be around dragons, see them as inspiring and deserving deference. When these ideas were instilled it was just four dragons gaming a human society, but over time connecting each generation of power-holders to those four dragons--nearness to dragons is good and right, being part dragon is better--that's going to ripple through society.
Half-dragons and dragons would likely be welcome in the society entirely independent of the power structure...but eventually their ubiquity would create implications for the power structure.
Think in terms of 33 generations: over that much time, new dragon clans emulating the structure of the old ones have time to develop and grow even if they're not acknowledged by the law or assigned power. And at the same time, the perverse incentives involved in the four dragon clans maintaining their lines means they're desperate to shore up their number of sorcerers and draconic sorcerers by any means necessary. Over time, these secondary clans begin to marry into the four main clans--by law they're kept subordinate and by kinship they're theoretically not-family, but even if the law says "people marrying into the four ruling clans don't have power" they can still have de facto power by being effective at ruling or just savvy at working the intrigue systems.
There's a lot of history of kings marrying women who nominally have no power in a patriarchal kingship, but become powerful because they're smart and their husband isn't...and a smart person with no formal qualifications is sometimes more important than having all the right pedigrees. There's also a lot of history of families providing a bride for a higher-status noble, then the entire groups elevates themselves by taking over the household of their noble in-law.
If your players want to crack this society open, or just play at intrigue...that really long timeline can pay dividends, because it means shameful secrets and hidden compromises that the ruling class conceal because it undermines the premise of their rule, as well as complex internal factions that know things about one another but are holding off on the high-society equivalent of Mutually Assured Destruction.
(At least one ruling dragon clan has no direct lineage from the dragon they claim as their ancestor. Given how royals roll, I 100% guarantee it.)