Well, being cynical, being known to be Darcy's close friend and a regular visitor at Pemberley, would do a lot to make someone as new money as Bingley acceptable to other old money types with a suspicion of arrivistes.
I mean, I don't think Bingley does think that for a minute, but he doesn't need to be a fortune hunter for him (and more obviously his unpleasant, snobbish sisters, who like to forget their family money is from trade) to benefit from the association.
Mrs Hurst doesn't seem to have married particularly well (not only is Mr Hurst a bore, but a man of 'more fashion than fortune'), so presumably Caroline Bingley wants to do better than her sister, and she's not just doing her brother a favour by acting as his housekeeper at Netherfield, she's in it partly for Darcy or, failing him, other upwardly mobile matches.
She's an unpleasant character, absolutely, but I assume some of her ill-concealed horror at the shenanigans of the Bennets and the Gardiners' Cheapside address is a desire to distance herself and her siblings from their 'trade' roots. And routinely performing that horror in front of their most blue-blooded friend is her way of saying 'See, I'm at your level!'
Lizzy is far more socially secure than she is. The daughter of a landed gentleman (despite the entail), it doesn't occur to her not to walk three miles cross country to see Jane when she's ill at Netherfield for fear of it might make her look declassée or countrified in front of the Bingleys.