P&P does state that Miss Bingley was to "keep [Bingley's] house" and "preside at his table." That's not a fan invention.
As to why Georgiana doesn't live at Pemberley primarily until her brother's marriage, maybe it's a matter of the lifestyle of an unmarried brother. Darcy is probably unimpeachable in that regard, but the same could not be said of Crawford. A married man, theoretically, is out of that wild oats phase of his life that would be inappropriate with an unmarried (and especially a very young) sister in residence.
As to why Georgiana doesn't live at Pemberley primarily until her brother's marriage, maybe it's a matter of the lifestyle of an unmarried brother. Darcy is probably unimpeachable in that regard, but the same could not be said of Crawford. A married man, theoretically, is out of that wild oats phase of his life that would be inappropriate with an unmarried (and especially a very young) sister in residence.