That's a fair counter. I hadn't considered the life of the family before Sachi's birth.
That being said, if we're talking about coincidence then Sachi's luck stat and everything about her is also a huge coincidence (until proven otherwise which it probably will be if we're being honest). Also, the disparity between a territory having their highest highs and the norm is going to be big no matter what. If no one could even figure out why the territory was doing so well in the first place, then it would be reasonable to assume that the marquisate would take greater risks to get back to the same level of prosperity as before and without the magical 999 Luck stat, they can actually fail (depending on the influence Sachi's power has on the world around her).
All of that to say we just be theory crafting.
It sounds to me like what you're describing just comes down to mismanagement of their realm. Every noble family will have to deal with ups and downs in their realm, so under normal circumstances, you'd expect people to invest and save during the good times so that you can weather the bad times. Regular bad luck shouldn't be able to bring down an otherwise prosperous and competent noble family. So that would make me believe that either they weren't competent or the bad luck wasn't normal in some way.
My point when I mentioned "huge coincidence", was that if it turns out that the noble family is just really bad at managing their realm, it would mean that them kicking out Sachi didn't directly cause their downfall. Which seems a bit at odds with how the story seemed to have framed the situation for me (though not impossible).
I mean, if a competent manager who hugely contributed to a company's success retires, it shouldn't be that former manager's fault for retiring if the company fails soon afterwards (assuming he didn't do things like sell company secrets after his retirement).