One is the lonliest number.
You'd expect the number of descendants to either diverge to a large fraction of the entire population, of disappear completely. This will be true of almost any way to describe "descendant", unless there is a serious feedback loop.
The hard part is avoiding extinction; I can give you a dozen feedback loops that keep the number from diverging. For example, the good old highlander trick, where people with the gene can detect each other and are compelled to kill each other for whatever reason.
The problem is that once the values are small, random flux means that there is a increasingly decent chance at extinction. 1 is a very small number.
We could go with feedback loops in each direction. Imagine the line has a reserve of magical luck. But this gets diluted when there are more of them, and the one with more luck actually imposes bad luck on the other members of the line in order to give more luck to its source.
The sole descendent thus has lots of luck, and survives. When there are more, the stronger bearer of the luck gene kills off the weaker members, unless they are focused on the survival of the stronger bearer. Luck remains concentrated, a single descendent is a given.
The "luck" could be something more sinister. Imagine if there is a demon who is watching over the line, and an order of assassins trying to thin it out. When the demon is distracted they succeed; when the demon is protecting one mortal, they don't stand a chance.
Another possibility is that the ritual requires a specific combination of bloodlines. What if there wasn't just one person in the ritual, but a set of them?
In order to complete the ritual, you need someone who is a descendant of all of them, as there is a limit on how many more people can join the ritual, and a requirement that every bloodline be present.
So the last remaining descendant could instead by a really careful breeding effort by the organization in question. They would have to find validated direct descendents of each of the bloodlines, perform specific costly rituals on them to "bring the blood back up", then breed them.
Such an effort would take literally generations of effort. Things would go wrong along the way.
Toss in a time based limitation (celestial alignment? 1000 year deadline?), and it might not be the sole descendant but rather the only viable candidate.