Another Chicagoan here. I feel that one can definitely live in Chicago in a very Mustachian manner with children. My wife and I have been married for 3 years and will probably have children within the next 4 years or so. Let me address the major gripes.
Housing: Yes, housing can be expensive, but this is not a given. One option that many mustachians can get behind is rental property. Chicago's great residential neighborhoods are chalk full of 2 and 3 flats that are wonderful options to be live in owners/landlords. My wife and I currently live in a condo, but are saving to be able to make a downpayment for a building like that for when we have children. These buildings usually have a small yard and a garage on the alley behind the buildings. The great part about the garages is that it can be an extra stream of income since parking is so desirable. You can either get a premium on your units if there is parking, use the spot yourself, or rent it out on its own.
Other costs of living (food, entertainment, etc.): This is where I think people have the biggest misconception about major cities. The earlier poster who mentioned the great ethnic food got bashed a little bit because eating out is so anti-mustachian, however, in Chicago at least, great ethnic food is usually a FRACTION of the cost of trendier restaurants and even fast casual type places (Chipotle etc.) and many of them are BYOB. This is a fantastic option to be social with those friends that always want to go out to dinner or drinks. My wife and I always steer the dinner plans to one of these ethnic restaurants and bring our own wine/beer, and take any leftovers...which we usually have, since portions are usually great. We also have some friends that don't want their own leftovers...guess what? We take those too and viola, we have lunch for almost an entire week. There are also a TON of free things to do in the city: Parks, Classes, Art Galleries, Museums, and not mention an amazing lakefront. The outdoor beauty of southern Wisconsin and Western Michigan are also short day trips away.
Schools: Another bit a a misnomer, but a legitimate concern for parents. This is extremely dependent on the neighborhood. In Chicago though, there are many very good neighborhood schools and plenty of public magnet programs, gifted programs, arts programs, etc. I have worked in 2 different schools in Chicago (Charter Schools, though no longer in that field) and my wife works in Chicago Public Schools. The schools I worked in and that my wife worked in were very diverse, had a lot of students from very low income families and also did not have great test scores...and I would have sent my own children to them in a heartbeat. Just because the school does not test through the roof, does not mean that the instruction is poor. This is just an area that parents have to be informed on and be proactive.
Transportation: One interesting point brought up, is that public transportation can be expensive. This is very true if you are using it daily and also own a car. That said, Chicago is a fairly bike friendly city, and walkability is also huge. Within half a mile from my condo are 7 parks, 2 standard grocery stores and countless ethnic grocery stores, a hardware store, an (subway)EL line that goes to the airport, 4 very useful bus routes, etc. Also, the person who said it would cost $2 in gas to get the airport...here in Illinois there are expensive tolls you would pay both ways that would negate the savings of taking the train. The train is $2.25 to get to the airport and $5 back into the city, so I would say the train saves money and I can read my book or the MMM blog on my way to and from the airport.
Other Benefits: I do think their is a social benefit to children to be exposed to all types of people and while I am sure their are suburban areas that do provide this, that is often not the case (I grew up in a lovely extremely homogeneous suburb about 1 hr west of Chicago). I think this is a social benefit for the child as well as society as a whole. And though Chicago is extremely segregated (which I could write an even longer post on), there is much day to day interaction with all different types of people within the city.
So, do I think living in a big city saves money? Probably not, but also don't think it costs more money either.
Would love to hear any comments on the above