These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Les Yay: Good God. Emma even says she's interested in Harriet because she's so pretty and passionately insists to Mr. Knightley that no man in the world is good enough for her. Not to mention the wording of Emma's relationship with her former governess.
Rescued from the Scrappy Heap: Emma again. Jane Austen wrote before she started that, "I am going to take a heroine whom no-one but myself will much like."
Values Dissonance: Emma's ignorance of Harriet's rank and insistence that the circumstances of Harriet's birth should not deprive her of the same respect and opportunities open to richer, less-tender-hearted women is portrayed as a bad thing. *
In at least one adaptation, Emma lets her imagination run wild and assumes that Harriet's mysterious father is a member of the royal family! This is why she thinks Harriet is suitable for Mr. Elton - not because rank doesn't matter, but because she's imagined a much more illustrious background for Harriet than is plausible. (Mr. Elton would realize that no royal bastard would be enrolled in a school designed to turn out governesses and ladies' maids, but Emma doesn't.)
More an 'unrealistic given the circumstances' thing. Also in-universe Values Dissonance where Harriet's obscurity isn't a problem to Emma but she considers association with Robert Martin demeaning. And another in-universe example in the great respect Miss Taylor/Mrs. Weston garners as a governess versus everyone being very sad about poor Jane having to become a governess.
That has more do with situation. Miss Taylor/Mrs. Weston lucked out as a governess, getting a good rich family that treated her like a member of the family and loving kids THEN marriage. Marriage is highly unlikely for a governess, which is the only way OUT of being a governess. If Miss Taylor was less beloved she would have been shipped out when Emma was sixteen. You get hints of this with Mrs. Elton being very snide about Mrs. Weston because of her background.
Knightley's objection is probably because it's not fair to delude the rather simple-minded Harriet into believing that she doesn't have a problem... in fact, more to the point, to do so while derailing her marriage to a yeoman with whom she would have been perfectly happy. (Even to the yeomanry it would have mattered, albeit not as much- Robert is still a catch, even if he does get his hands dirty.)
For a completely different example of Values Dissonance, it's eventually revealed that Mr. Knightley has been in love with Emma since she was thirteen...at which time he was twenty-nine!
Wangst: Miss Bates. Pull yourself together! (Notably the opinion of Emma herself, on her bad days.)
The Woobie: Miss Bates is pretty annoying for most of the book, but you want to give her a big hug after Emma insults her.