"Only the owner of Winterne can awake the winter magic"
"They awaken their winter magic by swearing an allegiance with the Winterne owner"
"People wth winter magic insictively swear allegiance with the owner of Winterne"
And why exactly are you special? First of all isn't clear at all if i is linked by bloodline or by role. Both have issues, as the first would be applyed for every descendant of the family, the second, why being in a role allow you to have this kind of ability?
Considering that for what we saw magic doesn't seem bloodline dependant. (as in case of Athy and her sister, where she have dark magic while Lillian light magic, and if this was an exception people would have rumored hard on that, it's basically a free pass to imply that the child is fruit of a betrayal, or not being an actual blood relative).
A system should however respect a predefined set of rule and the rules should be choerent and explained (or at least inferrable) . Once the system is built upon there can be extended, but not turned (unless the turning of the rules is a proper plot point).
My biggest grievance with magic system based on bloodlines is a partial inchoerency: most of these systems have the magical bloodlines as nobles (often high ranking ones), wer are shown that these families have members with a varying degree of talents, and ok I'm fine. But in reality nobles would make a lot of childrens, as spares.Yes they would be sent to monasteries or knight orders (or in case of fantasy stories, a magitians' order), some would be sent to other families to forge alliances. But they won't disappear. They would bear childrens probably, and these children should have the magic abilities proper of the family bloodline. Instead most of these stories make the magic limited to the main group of the family.
Also they never explain how the thing started, making seems like that a family as a specific power from the start of the world. Note, discovering the origin of one powers can be a plot point (and imho an isekai with this kind of story can be better then the 90% of the isekai already made), if not something that allow me to understand how it happened. Example unrelated to asian stories: In Harry Potter, it is never stated how mages come to be, we know however that family with no history of magic, sometimes (with a very small probability) have a magical child (called as "mudbloods"), and that this wasn't a new pehnomenon at all. We can infer that the first mages started exactly in this way, and that even longstanding pureblood families have at least one mudblood ancestor.
We don't know if this theory is correct but it is logic and sound, as it does explain how everything started, and it doesn't violate any rules of the system.
True we can make tghe same theory in a case like this, but what's missing in this case is the element on which we can start the deductive process.
Similar critcism to system that use caste discriminant can be made (for example nobility). This has also the issue that this kind of system seems to consider nobility as a static caste. Now generally speaking is true, but nobles can lose their statuses and non nobles can have assigned a noble rank. Maybe it has more sense in japanese views (as all japanese nobles had imperial blood, now if this is true or the idea is born as propaganda, or both, we can't say), but not in an european nobilty style.
Considering then the stories where there are this kind of system in place but then there is an exception becouse who knows.
Sorry for the rant