Chicago's hypercentralized layout may play a part there, too - the downtown area where tourists go is really just for tourists, and to some extent it's particularly for tourists making weekend trips from the suburbs. With the possible exception of Chinatown, all those great neighborhoods you're mentioning are pretty far off the beaten path for a visitor.
That $2k-ish per month is entirely eaten up by the cost of housing. Median rent for a 2BR in SF is $4,000/mo. Chicago is a little over half that. Not to mention much lower taxes than in California.
To give perspective in Chicago $4k a month can you get a modern 3,000 sqft home or the penthouse in a swanky building vs a studio/ 1br in SF.
I myself don't like the Bay Area for my own perception of the lack of diversity there, but again I would not be surprised if that is my bias showing through.
(and for instance I live in a neighborhood that has a history of multi-ethnic families that goes back to before that was socially acceptable nearly anywhere else in America)
San Francisco for instance is segregated pretty dramatically with the Asian population (and a mixture of Latinos) pushed to the western edge and the white population centrally located. Not to mention the elephant in the room of Oakland.
When we start talking about diversity of other sorts such as income, education, nationality etc the picture is similar.
That is to say, my experience of San Francisco was one of homogeny, the opposite of my experience of Chicago. But I've experienced Chicago more broadly as well, so as I say I'd like some numbers to backup the assertion that San Francisco is more welcoming of inter racial families.