I do agree the phrase conveys information about both the Bingleys' family and source of their fortune. Yet I am not at all convinced the Bingleys are from a gentry family or the rule of primogeniture applies to the Bingleys.
In England, traditionally the nobility and gentry were landowners; the Bingleys were not landowners.
If there was any Bingley family estate in the north, I'm sure Caroline would mention it to Lizzy or Jane. ;-)
The narrator says Bingley's sisters were 'very anxious for him to have an estate of his own'. ch.4.
A family estate would be a social markup for the Bingleys.
'Charles Bingley is a gentleman of leisure, and already associates with such a prestigious member of the country gentry as Darcy. Yet he is new money 'acquired by trade' and we see him buying his way into the gentry class.'
( from The Cambridge Companion to Jane Austen ed. by E. Copeland, essay on 'Class' Juliet McMaster.)
As the three professions for younger sons of gentry were the Army, the Law and the Clergy I doubt Bingley, snr was the younger son of a gentleman who engaged in trade.
Unlike the established gentry families in the book, if there are any genteel relations of the Bingleys we never hear about them.
[ I'm afraid I don't believe because we don't hear of genteel relatives, they must be there. ;-) ]
Louisa's husband Mr Hurst, ' a man of more fashion than fortune' is the only genteel family connection we hear of.
Certainly the Binglettes choose to forget where their money came from. The circumstance uppermost in their minds is they are of a respectable family. Yet they are status-hungry, 'proud and conceited'. Caroline Bingley is over-eager to ally herself and her brother with the Darcy family.
In S&S, Mrs Jennings was the widow of a wealthy tradesman whose daughters married into gentry, indeed one married a baronet.
In JA's society it was more acceptable for wealthy tradespeople like the Bingleys to be accepted into the gentry class, rather than gentry to earn money by trade.