Bill has always been a proponent of elevating the perception of the African American household and family. I think he did a great job providing an aspirational ideal.
In a sinister way, I think white people liked it because his portrayal of the black AA family most closely resembled a white middle-class ideal as well. Self-imposed white guilt allowed white middle-classes to "get in touch with blacks" without really having to actually get in touch with reality. But that's TV...never reality.
Let's not forget, too, that Bill produced and aired the show during his hey-day as a comedian/actor. He was the cock of the walk in those days and at one point, was the highest paid entertainer, period.
The show had an excellent sense of humor too, and Bill provided everyone with an ideal on how the father figure in any family would like to perceive himself (and how women wished their husbands would act), which allowed him to cross several demographics. Standard family problems were tackled in a friendly way and wrapped up neatly in 22 minutes.
I doubt folks would tune in to a folksy sitcom these days in any serious numbers. The closest I see today is a show called "the Middle"...yes it's white, but it borrows from the plotlines of the Cosby show.
I really liked what he did with the show. He does not get enough credit today.
I agree with pretty much everything you said there, ...but I kind of take issue with the "aspirational ideal" part.
Perhaps the Cosby show was merely portraying a slice of life for a particular demographic. A group that is often never seen or heard of, but exist in homes all across America, ...a middle class Black nuclear family.
Shows like Family Ties portrayed one demographic, All in the Family portrayed another demographic, and RoseAnne portrayed yet another. All exist within the wide spectrum of American families. Just because Cosby differed from Good Times or Sanford & Son, or whatever stereotype or traditional depiction of Black family life may have been prevalently portrayed before, doesn''t mean it was "aspirational"