despite Bingley's appearance as a most eligible bachelor, what is revealed about his family, his age, and his inexperience
It is quite agreeable to admit that Charles Bingley may appear to the naked eye to be "the" most eligible bachelor in Pride and Prejudice but, in reality, he is merely "one of the better ones".
In the surface we find that Mr. Bingley is quite handsome, amiable, laid back, and let's not forget: quite rich. However, a good pre-Victorian marriage is not only a contract to ensure financial stability, but it is also a way to enter the bloodlines of good and prestigious family names and social ranks.
This latter would have not been possible to acquire through Bingley. We learn in the story that, although he is friends with the fashionably aristocratic set of Fitzwilliam Darcy, Bingley's family acquired their money through the trade business. This is the modern equivalent of comparing a Prince of the blood to a Nouveau Riche guy. The level of social ranking are comparatively different no matter how much money is involved.
Furthermore, we see that Bingley allows Darcy to basically run him. He listens and accepts every word spoken by Darcy, and even leaves Jane behind in favor of Darcy's opinion that she is not good enough for Bingley. This, and the fact that Bingley not once makes a formal move to demonstrate a special kind of love for Jane, make up for a very inexperienced and annoying man.
Finally we know that he does make enough money but Darcy still makes twice as much. So if we gather all the facts and look at things objectively, it is Darcy and not Bingley who would be deemed as the most eligible bachelor: He may not have the best personality but he has a family name, a lot more money than Bingley and, to a point, seems passionate enough to declare his love to Elizabeth in one of the most bipolar and odd marriage proposals in literature.