I was scrolling through the Korean wiki of the Natsuyu characters and I came across this in Matoba's profile. Korean wikis contain a lot more discourse and theories between fans in contrast to the English Fandom wikis which only state Canon information hahahahaha. The google translation is a bit wonky but what's your opinion on this?
hmmmmm ok i had to look up the webpage itself and see what chapter they're referencing for that "nuance that the head position is not passed down through lineage" turns out the little [6] citation mark there doesn't actually cite any chapter, but just more explanation about the dangerous nature of the exorcist profession and that spiritual power aren't guaranteed to be inherited from parent to child, etc. which i don't consider to be a valid supporting evidence to what they're saying about matoba clan inheritance system. as for seiji saying "the previous head", since the page doesn't specify what chapter that line is from, and i can't remember off the top of my head where seiji specifically said "the previous head", so i'll talk about instances i remember of seiji talking about other matoba clan heads:
-in a hand extended, seiji said "...or so the current clan head says": he uses the title because that gives weight to the opinion that the exorcism industry is on the decline, since it's the stance of the very person at the top of the industry, and it wouldn't land the same way if he said "my dad/relative says..."
-in miharu arc, "i'm disappointed in the matoba leader of that time": the miharu family died out before the last three pillar ceremony that welcomed lady white claw/shirotsume-no-kimi 40 years ago, and considering matoba's assumption that he will no longer be the clan head by the next ceremony in 30 years time (which yes as we all know seiji is passively suicidal but he's also a stone cold pragmatist and i've always read that line to be informed more by the history of exorcist being a dangerous profession and matoba clan heads being an even more hazardous job and so they tend to have rather short tenures) it would've been a matoba clan head from 2 generations (or more) ago that was in office and therefore not seiji's parent or someone from their generation.
↳if you or anyone reading this knows in what chapter seiji said "the previous head" please let me know in the reply and i'll look into it
afaik (again, off the top of my head), there is so far no concrete evidence to prove or disprove what they're saying here about matoba clan leader's position being a free for all to anyone in the entire clan. BUT, i'm more inclined to believe that while it's not impossible for the position to go to someone not directly related to the main matoba family, the children with direct lineage have precedence over those from other families in the clan. if none of the children of the entire matoba family in the next generation is deemed good enough (most likely in the sense that none of them have strong enough spiritual power + aren't skilled enough), then they might adopt a children from another family within the clan and name them heir.
in a world unbent, while seiji did only say "i'm here on the clan's orders", takuma then specifically criticized "matoba-san" for said order. seiji's phrasing insinuates that the order came from the leader, so takuma's words next would imply that said leader is seiji's parent or at least a relative.
↳ side note, the official translation of the manga said "my family told me to come" and "mr. matoba again", which on the surface supports my arguments, but it must be clarified that in jp seiji says "the clan (ichimon)", and "matoba-san" (which does not denote gender, hence why some ppl headcanon that the previous clan head was seiji's mother instead). just mentioning this for transparency.
chapter 123 also shows that the only children being considered for heir is the 2 matoba siblings. again, could be that the children from other families in the clan are significantly weaker in power, but a significant theme in auction arc is blood & inheritance (esp in the eyes of youkais) so i think the matoba siblings & discussions about them being heir candidates being brought up in this arc means something.