I'm not going to comment on what makes a romance a romance, but I do have some comments on Austen's novels. I've made these points before - I think more than once - but at least someone will not have seen them. Four points, then four discussions:
1. Austen's novels are definitely romances, but only on the surface.
2. I think every one of the six novels is a bildungsroman, a coming-of-age story.
3. They are absolutely, positively satire.
4. They are social commentary, increasingly so.
1. It's not a spoiler to say that every single protagonist ends up with the guy (a) she wants at the start, or (b) figures out along the way that she wants. It's a happy ending because she gets who she wants. Wikipedia even includes Austen as someone who developed the romance novel. I think, however, that Austen used the romantic form for other purposes, because she was a woman in an age when it was difficult to do anything else. Women did write in other genres (e.g., Frankenstein) at the time, but it was really rare. Sorry if that qualifies as misogynistic, as discussed above (it's not intended to be, but the road to hell ... you know), but I really think she was mostly about other things. I don't think this idea is mine - I'm probably parroting someone else, but I can't remember. It also might be wrong, but I just don't think Austen was primarily interested in the romance as a form - for instance, I agree with what OP said above about Emma and Mansfield Park; I think it applies to some others.
2. Every heroine except Elinor Dashwood undergoes some sort of disillusionment. Elizabeth Bennet, Catherine Morland, Emma Woodhouse, and Marianne Dashwood are disillusioned about themselves. Anne Elliot and Fanny Price are disillusioned about others (Lady Russell and family of origin in Portsmouth, respectively). Anne's experience occurs years before the novel starts, and is discussed just after the beginning and at the end; Fanny's occurs late in the novel. The first four above are embarrassed and ashamed of themselves to various degrees once reality intrudes on their illusions about themselves. For instance, after she reads Darcy's letter a second time, Elizabeth Bennet says, "Until this moment, I never knew myself." Only Elinor Dashwood seems to have been born disillusioned - about her mother, her sister, their half-brother, his wife, her brother, Lucy Stone, and herself. Parentified indeed! In contrast, Emma's and Marianne's self-disillusionments are particularly brutal for them, while Fanny gets off easy in this way (definitely not in others).
This, to me, puts all the novels in the genre of bildungsroman, which Wikipedia defines as "a literary genre that focuses on the psychological and moral growth and change of the protagonist from childhood to adulthood." (Wikipedia also says that some who study bildungsromans think Pride and Prejudice doesn't qualify. Well, I disagree.) I find that, as someone who has been mentioning this repeatedly here on Tumblr (and once on Reddit), it's interesting that the fandom almost never mentions this or discusses it. Why? It's there in every novel! Why don't Austen fans, who are mostly women, talk about the psychosocial development of young women in these novels? It's especially baffling about fans who are beyond this age, who must have experienced the normal sorts of disillusionments that life gives. As a man, I've certainly dealt with my own repeated self-disillusionments; I think it's normal for most people of any gender.
3. I don't think the satirical aspects of Austen's works need much discussion. They're there in every single novel, popping up over and over, sometimes in action, sometimes in the author's commentary, and they are hilarious.
4. Someone said here on Tumblr a year or two ago (was it OP? maybe it was - not sure) that if one follows the dates of publication of Austen's novels, one sees that they increasingly criticize the social structure of British society. (Not class! That's a Marxist term from 30 years later. It's rank.) It was a great post and I am sorry that I cannot find it. It went through all six novels, showing how Austen's criticisms grew. Those born to their positions in her early novels seem to "deserve" them, while her last book, Persuasion, presents an idiot baronet who is entirely self-satisfied, pompous, superficial, spendthrift, and in love with not only his own visage (six mirrors), he's in love with his own unearned social position (e.g., his repeated reading of The Baronetcy). In contrast, navy men earn their social positions through sheer competence, and totally deserve them (Admiral Croft, and Captains Wentworth, Harville, and Benwick).
Famously, both onions and ogres have layers. Jane Austen's novels all have many layers, as I hope that I have shown. Are they romances? Are they satire? Are they other things? Yes, yes, and yes. Maybe that's why people keep going over this issue. They're a lot of things. She was a genius. That's why we're still talking about her work, two centuries later, and counting.
And if you think I'm wrong about anything here ... okay. Let me know.