I have no clue what you are trying to get at. Nothing in those pages says anything about why they have arranged marriages. Arranged marriages are not even brought up until the very last line.
Given how the two of them talk about the whole thing, they're totally resigned to being betrothed to people other than the one they love. That's arranged marriage right in the face of it, and it's somewhat difficult to see why their families would brother with the whole thing if there weren't practical (ie. economic and strategic) reasons to. And from Sayuri's comment, plus the presumably solidly middle-class Kohane's bewilderment, we know that's something particular to and not uncommon for wealthy upper-class families.
Ie. political marriages between powerful clans.
The stumbling block was literally put in place because of the sexes. Daughters couldnt inherit so they would be wed into another house to form an alliance. Families of only daughters were doomed to die unless they got lucky and were able to adopt the second or third son of another house. If daughters can inherit then who gets adopted into which family? Who gets the inheritence? Its a system built around the patriarchy, remove the patriarch and the system fails.
You're forgetting that the marriage itself forms a certain bond between the two families. This was often the primary purpose of historical political marriages, the whole inheritance shebang often being somewhat secondary or causing problems further down the road (due to inconveniently inherited claims) - or in some contexts all but irrelevant depending on how the relevant laws and customs happened to work.
It's alliance building by the admittedly somewhat primitive but relatively effective expedient of binding the families together by means of marriage and, eventually, children. (Actual track record of those ties has been... patchy to say at least, but practical substitutes have been few and far between.)