So far all was perfectly right; but Lady Russell was almost startled by the wrong of one part of the Kellynch Hall plan, when it burst on her, which was, Mrs Clay's being engaged to go to Bath with Sir Walter and Elizabeth, as a most important and valuable assistant to the latter in all the business before her. Lady Russell was extremely sorry that such a measure should have been resorted to at all, wondered, grieved, and feared; and the affront it contained to Anne, in Mrs Clay's being of so much use, while Anne could be of none, was a very sore aggravation. (5.11)
It's just incidental to Lady Russell's objection that Anne is more competent and reliable than Mrs. Clay – what really burns Lady R.'s pastry is that Elizabeth is choosing someone outside the family.
She acknowledged it to be very fitting, that every little social commonwealth should dictate its own matters of discourse; and hoped, ere long, to become a not unworthy member of the one she was now transplanted into. With the prospect of spending at least two months at Uppercross, it was highly incumbent on her to clothe her imagination, her memory, and all her ideas in as much of Uppercross as possible. (6.3)
The family is like society in miniature, and it's here suggested that, to be a good family citizen, one must conform, to the extent of becoming almost a different person (because between "imagination," "memory," and "ideas," that's pretty much a whole identity right there). Though Anne thinks this at the beginning of her stay, it would be interesting to look more closely at how she lives up to her initial plan.
"You know," said she, "I cannot think him at all a fit match for Henrietta; and considering the alliances which the Musgroves have made, she has no right to throw herself away. I do not think any young woman has a right to make a choice that may be disagreeable and inconvenient to the principal part of her family, and be giving bad connections to those who have not been used to them. And, pray, who is Charles Hayter? Nothing but a country curate. A most improper match for Miss Musgrove of Uppercross." (9.13)
"Think of the family!" is the "think of the children!" of this novel – the statement characters can use to shoot down any plan they don't like. Here Mary uses it to defend her selfishness in putting her own snobbery ahead of Henrietta's marital happiness.
Louisa seemed the principal arranger of the plan; and, as she went a little way with them, down the hill, still talking to Henrietta, Mary took the opportunity of looking scornfully around her, and saying to Captain Wentworth--
"It is very unpleasant, having such connexions! But, I assure you, I have never been in the house above twice in my life."
She received no other answer, than an artificial, assenting smile, followed by a contemptuous glance, as he turned away, which Anne perfectly knew the meaning of. (10.18-20)
Mary's tendency to judge others by their family connections means that she lives in fear of others judging her the same way – she can't imagine that her snobbishness could put another person off even more than her low-class relations.
In Lady Russell's view, it was perfectly natural that Mr Elliot, at a mature time of life, should feel it a most desirable object, and what would very generally recommend him among all sensible people, to be on good terms with the head of his family; the simplest process in the world of time upon a head naturally clear, and only erring in the heyday of youth. (16.7)
Lady Russell sees recognizing the family hierarchy as a "natural" thing to do. In her eyes, paying respects to the head of the family is not something Mr. Elliot needs any other motives for, but what any thinking person would do, because it's the law of nature. Although of course calling something a law of nature doesn't mean it actually is one.
For a few moments her imagination and her heart were bewitched. The idea of becoming what her mother had been; of having the precious name of "Lady Elliot" first revived in herself; of being restored to Kellynch, calling it her home again, her home for ever, was a charm which she could not immediately resist. (17.25)
This passage shows that Anne does have family feeling, it's just very different from the feelings of her sisters and father – for her, family is about home and domesticity, and is linked to her mother, who seems all but forgotten by everyone else. It's interesting that, for most of the other characters, marriage is all about whom you get connected to, while Anne actually thinks about making a home for a new family unit.
Even the smooth surface of family-union seems worth preserving, though there may be nothing durable beneath. (21.52)
Many of the characters feel this way – why is keeping up family appearances so important?
"Such excellent parents as Mr and Mrs Musgrove," exclaimed Anne, "should be happy in their children's marriages. They do everything to confer happiness, I am sure. What a blessing to young people to be in such hands! Your father and mother seem so totally free from all those ambitious feelings which have led to so much misconduct and misery, both in young and old." (22.24)
Anne points out the usually negative tendency for parents to try to live through their children, only to hold up the Musgroves as an example of how this approach can make everyone involved happy since they have their children's best interests at heart.
They found Mrs Musgrove and her daughter within, and by themselves, and Anne had the kindest welcome from each. [...] It was a heartiness, and a warmth, and a sincerity which Anne delighted in the more, from the sad want of such blessings at home. She was entreated to give them as much of her time as possible, invited for every day and all day long, or rather claimed as part of the family; and, in return, she naturally fell into all her wonted ways of attention and assistance, and on Charles's leaving them together, was listening to Mrs Musgrove's history of Louisa, and to Henrietta's of herself, giving opinions on business, and recommendations to shops; with intervals of every help which Mary required, from altering her ribbon to settling her accounts; from finding her keys, and assorting her trinkets, to trying to convince her that she was not ill-used by anybody; which Mary, well amused as she generally was, in her station at a window overlooking the entrance to the Pump Room, could not but have her moments of imagining. (22.32)
While Mrs. Musgrove and Henrietta do seem to have more genuine affection for Anne than her own family, Anne still gets stuck listening to everyone talk about themselves and helping others out.
Anne, satisfied at a very early period of Lady Russell's meaning to love Captain Wentworth as she ought, had no other alloy to the happiness of her prospects than what arose from the consciousness of having no relations to bestow on him which a man of sense could value. There she felt her own inferiority very keenly. The disproportion in their fortune was nothing; it did not give her a moment's regret; but to have no family to receive and estimate him properly, nothing of respectability, of harmony, of good will to offer in return for all the worth and all the prompt welcome which met her in his brothers and sisters, was a source of as lively pain as her mind could well be sensible of under circumstances of otherwise strong felicity. (24.10)
This passage sets up a new set of family values to replace those that Anne's family believes in: instead of Anne's family being superior to Wentworth's because of their class, they're inferior because they're jerks. Family here becomes simply the people that are in it, and if those people aren't cool, then the family as a whole is no good, no matter how aristocratic they are.