People are I think making the mistake of not accounting for how an aristocratic society actually operates. Sandra is not betraying Bernadetta, she is glad for her happiness, and thinking of it in the long term.
A woman can take headship of a family for one or multiple generations, but she has to be the daughter of the paternal (meaning the male one) if her own children inherit unless her husband is from a branch family related to that line (as was very typically the case in such circumstances but isn't in this one) then the title of nobility will either be usurped by the husbands family (irrespective of the terms of marriage, this happens in fact even in cases of morganatic marriage, which is why if they could the family would try to arrange on with a branch family, and even that was controversial depending on how far back they had diverged, also this usually had immense religious significance outside of Christian societies where ancestor veneration persisted) or reverts to the crown depending on the laws in question.
Bernadetta has got what she desires, she shall be right hand to the future Queen. Under such circumstances family headship would be convenient in order to secure her position and to use her family for leverage among the nobility (she would be the one negotiating their associations and marriages as family head), however she would be expected to adopt a nephew as heir in order to continue the lineage.
The issue is that she intends to marry, her husband is a man of considerable station and her children will likely inherit her excellence (the source of her brother's desire that his own children match it), Sandra wishes that a future conflict over succession is avoided, that her brother cleanly inherits the house and Bernadetta is free to live without the danger of a serious internal dispute (she is high nobility, her family will have many branch families of lower nobility associated with the paternal line, if the ruling family changes they will be left out in the cold so even the possibility will be an extremely volatile situation). Sandra also knows that if she doesn't inherit the house her proximity to the crown and the bloodline of her husband mean that she will likely found her own noble house anyway and receive her own titles. That would benefit her old house through association as well as they would be natural allies within the existing political\familial order.
Given the disputes between the crown and Church this is by far the best outcome. Sandra is entirely right to wish it, as ever she wears her heart on her sleeve, while the others are likely laying the groundwork for this.
Aini is acting like a knight errant, I think errant has clear enough connotations in English for people to understand why that may be frowned upon no matter the virtue of her cause, especially by characters committed to dynastic duty and the obligations of nobility (which she technically is, furthering the exasperation).
Orva is likely being subjected to eugenics, Erik's head maid is almost certainly of high birth, as the servants of royalty were, but likely of the lower nobility rather than the station for a direct retainer. She is physically robust and most likely of dutiful temperament, the hope is to breed out Orva's frivolity and his brother's frailty (likely why he didn't inherit the house in the end, their purpose is to provide knights to the royal family, though if the children took after Sandra that would likely work out just fine), in order for the family to fulfil their position without the need for direct royal intervention in family affairs into the future.
As is the point of the whole story the characters are responsible for considerable power and influence, it would be to the detriment of the whole for them to act based on personal inclinations. Orva was the nail that stuck out, he was hammered down, Sandra is rather loose tongued but everyone is seeking to guide high society down a harmonious path for the sake of the nation, she expresses that will in words, the others in the use of their authority.
The point is not to pick sides with favourites, but with the interests of the whole, this is what marked this story out from others, the depiction of relationships within such constraints.
