Sorry to highjack your post with an essay, but there's actually a common misconception here that I really want to breakdown.
One of the things that it isn't easy to notice these days is that Jane and Bingley actually are a commentary on love and society in exactly the same way the other couples are. It just isn't as obvious because the expectations and discussion over how people are meant to behave when in love has vastly changed in two-hundred years.
Jane exemplifies a common standard for young gentlewomen of that era: be demure (but never cold), friendly (but not too friendly), reserved about your true emotions (but always pleasing to everyone), appear grateful for every civil interaction a gentleman offers you (but never seeking or desperate for them), etc. She's beautiful, yes, and unfailingly kind, but her 'perfection' for contemporary readers would've gone far beyond that.
Because in many ways, Jane is the perfect gentlewoman. All those impossible virtues of good sense and perfect goodness and eternal gratitude and elegant grace are united in her. And in the Jane and Bingley love story Austen asks the question of how that behaviour, however generally admirable, can function in reality and then explores some of the drawbacks.
We actually see Charlotte allude to this directly in chapter 6. When Lizzy is happy that "Jane united, with great strength of feeling, a composure of temper and a uniform cheerfulness of manner which would guard her from the suspicions of the impertinent," Charlotte famously rebuts:
"It may perhaps be pleasant," replied Charlotte, "to be able to impose on the public in such a case; but it is sometimes a disadvantage to be so very guarded. If a woman conceals her affection with the same skill from the object of it, she may lose the opportunity of fixing him; and it will then be but poor consolation to believe the world equally in the dark. There is so much of gratitude or vanity in almost every attachment, that it is not safe to leave any to itself. We can all begin freely—a slight preference is natural enough; but there are very few of us who have heart enough to be really in love without encouragement. In nine cases out of ten a women had better show more affection than she feels. Bingley likes your sister undoubtedly; but he may never do more than like her, if she does not help him on."
This exchange isn't just iconic (and, in my opinion, a mark of Austen's genius for all it conveyed), it's a debate about society and its ideals vs the reality in practice. Since society has changed readers tend to see it purely as a commentary on Jane/ justification for why Darcy interpreted her the way he did/ foreshadowing for Charlotte's own choice, but it wasn't only that. It was calling out some downsides to women being perfectly composed at all times when the man they're in love with is a decent guy who cares about things like 'whether his affections are welcomed' and isn't so self-centred as to not have doubts over how someone who doesn't reveal much might actually feel. It's actually a testament to Bingley's character and general concern for others that he doesn't just assume that 'of course she likes me, she's polite and friendly to me,' when doubts are raised. You know who wouldn't have doubts? Arrogant and self-centred people whose priorities aren't others and think only about what they want. Though not directly said in the text, the Jane and Bingley temporary break-up does call into question whether behaving in this admirable way might actually push away the most considerate and thoughtful suitors.
And though I know modern readers are very prone to judging Bingley harshly for not returning quickly to Jane, keep in mind we live over two centuries later in a far more individual-focused society with different values. In the text Lizzy, who we all know has no qualms about being angry at others, ceases to be mad at Bingley almost as soon as she receives Darcy's explanation. He's not condemned by either her or the text for being persuaded that Jane was indifferent to him, and Lizzy actually comes to believe it's understandable.
I think another thing we've lost with the passage of time is just how bad the Bennets could be seen as. While Mr Bennet lives they're rich, top 0.2% rich for England in that era, and yet the daughters will have next to nothing for their class/upbringing and weren't taught many of the housekeeping/economic skills they'd need for a realistic future. I've talked more in depth about what they should have been saving according to contemporary accounts and done some maths here and here but the gist is they should've easily been six times as rich as they are. Let's not forget the lack of education too. I said it in one of those posts, and I'll say it again, if you knew a top 1% family who were constantly flirting with bankruptcy and 2/5 of their children were barely educated you wouldn't be wrong for thinking there were some serious problems in that family. Then there's the social vulgarity/silliness, but that translates much better to modern audiences so I won't go into that anymore than to say that decorum was a BIG DEAL back then and who you were 'connected with' could very literally affect your standing in society. Darcy and Bingley's sister's were snobbier about it than they should've been, but the core reasons for concern were actually valid. Even Lizzy very quickly saw the justice in Darcy's logic once presented with the facts so bluntly.
Bingley noticed these things, as everyone sensible did, but he's just too generous a person for that to matter enough to stop him from wanting to marry Jane. It was only being persuaded that she genuinely was indifferent to him that made him put aside his hopes.
We should also keep in mind that it wasn't just randoms who were doing the persuading, it was Bingley's best friend (who is used to believing himself an authority on others - a flaw he has to overcome in the course of the novel) and his sisters (whom everyone considered close friends of Jane and who would've seen her more than Bingley). Their motives were jaded by prejudice but for many contemporary readers these would've been the most reliable advisors anyone could have in matters like this.
Given the delicacy of the subject it's not like he could directly ask Jane herself until the actual proposal, or even begin acting more markedly and hope she responds in kind (the impropriety of which is similar to what we see with Marianna and Willoughby in Sense and Sensibility). Even when Lizzy knows Bingley liked Jane, knows that Jane still feels the same and suspecting that he does too, she doesn't so much as think about giving him a hint when she sees him again in Derbyshire. It simply wouldn't be proper, it's up to his intimates to speak with him about it. So, if Bingley wanted an outside opinion Darcy and his sisters were it; and, on paper, they're very good advisors on the topic of whether Jane liked him.
In most situations it would be a massive character flaw to think 'I don't care what all my closest family/friends/her friends say, I'm going to persist in thinking this girl likes me against their advice.' Keep in mind they knew each other for six weeks and he's never even been alone with Jane. His sisters have though. There's also a commentary in there on the moral pitfalls of influencing someone at all (which is explored in far more depth in Persuasion) but Bingley is never called wrong by the text or characters for not jumping to the assumption that his friend's being an arrogant snob and his sisters are bitchy snobs. A rich man who recognises he can be wrong is a good quality even today, and if we think in contemporary terms (and remember he's only 22) I don't think it's at all unreasonable that he was persuaded.
Which brings us to his whole personality: Bingley is in many ways a perfect gentleman socially. Charming and civil to everyone, uniformly good-tempered, and other than offending one or two young ladies by not asking them to dance, commits no social sins. He's also praised for being friendly and obliging - the latter being another trait which, as Jane Austen does with Jane's praised traits, gets explored via its weaknesses. Arguably the novel is one long exploration of the weaknesses of various traits, most notably those in its title, but this is already too long for that tangent.
Bingley's also very new money. Outright called the first gentleman (remembering that that word meant something very specific about education, dress, behaviour, poise, etc in that era compared to today) of his family, and his father was in trade. In a time where the middle merchant class was still establishing itself as worthy of being treated with respect by their 'betters' (and the mere fact of Darcy's close friendship with Bingley is the first clue that he's not as arrogant and snobby as Lizzy believes) his perfect upholding of an amiable ideal is a commentary in itself. Especially when we see Lady Catherine and Darcy, with their impeccable bloodlines, commit social faults arguably equal/worse to Mrs Bennet (herself not born into the gentry class and a negative example of social mobility to contrast Bingley's positive example) and Mr Collins. The highborn character who does embody appropriate social graces, Colonel Fitzwilliam, is interestingly not landed himself and needs an occupation.
Modern readers, without such a class based society which focused on social graces, are also less understanding of that 'obliging' aspect of Bingley's personality. But this was a time when, generally speaking, the richer and more important you are the more likely you are to get what you want and everyone else fell into line. It was so common that it wasn't even really critiqued heavily by Austen, some people were rich and had the means to do as they wished through money or social credit, and others followed if they wanted to be involved at all. We see this casually mentioned when Colonel Fitzwilliam says "I am at [Darcy's] disposal. He arranges the business just as he pleases;" which also helps us understand that the Colonel probably didn't have the income to own his own carriage or easily rent one to travel (which was EXPENSIVE). That context, of rich men not only ruling the world but also getting to decide what other people (in the Darcy/Colonel Fitzwilliam case, even older and higher-born people - and Bingley was younger and new to the gentry) do in their leisure time through virtue of their wealth, is the context we need to view Bingley in. Though Darcy was undoubtedly more important Bingley was still 2-2.5x richer than Mr Bennet and thus everyone else in the neighbourhood excepting his friend - and yet far from being the standard rich man who began dictating the social scene and choosing what to do without consideration for others, he was obliging. He matched what others were doing, had consideration for them, participated as though grateful to be invited instead of entitled to it. His obliging nature is part of what sets him up as a true gentleman and far more worthy than others who only adopt some of the social graces and miss how it's meant to apply to their whole character.
His personality is actually a very interesting study in what makes a gentleman a gentleman, and argues that the real qualities which matter have nothing at all to do with connections or family history. It's also an analysis of what obliging personalities can fall victim to, even when they're sensible, as Bingley is said to be. His whole character ties in directly (as does Wickham's more overtly) with Darcy and Lizzy's own journeys with true gentlemanlike behaviour and character. It's just not in a way which is at all easily noticeable to modern eyes without a background understanding of the society he functioned in, nor is it something directly depicted in the adaptions.
Anyway, sorry for the hastily typed essay and I hope I've convinced you that Bingley and Jane are an exploration of love and society just as the other couples are, and also a rather pointed social commentary on behavioural standards and changing class lines through social mobility. For all that Jane Austen's writing feels comforting and sometimes quite verbose, she actually fit an immense amount of commentary and meaning into every aspect of her books. Jane and Bingley are absolutely no different.
