As a kindergarten teacher in CPS....
I would never send my white child to a public, magnet school, etc. Almost any school at elementary age run by CPS. Race is an issue at public schools. If you are the only white kid at the school, the other black/hispanic kids will let you know. At my school, we have had 1 white kid every year. But, they only last one year. The white kid is endlessly picked on for being white. Rough environment to be in. Plus, you're child learns stupid-inner-city-neglected-child-lingo to try and fit in. Nothing like hearing kids endlessly say "duh" instead of "the" or "dat" instead of "that".
Logan Square is a party neighborhood. Hipster central. I would never let my child be exposed to that. He might grow up like that.... Lincoln Square would probably actually be good. Ask LisaLisa, she's raising a kid, but I think she and her husband live in the suburbs.
Beverly would be good. But way fucking far out there. Practically the suburbs. Hyde Park is actually probably the best place to raise your kid, in the city. Ray Elementary in Hyde Park is an exception to my above "rule". Good diversity and a good school. I don't know, as the Smith Westerns, they all grew up in the City. Good Public schools do exist in Chicago, but they are few and far between. You have to do your research and then you have to fight like hell to get your child in.