It's always a bit ironic to hear non-city realtors discussing Chicago real estate. Because they aren't in the city it's understandable that they don't know that one of the lovely assets of Lincoln Square is the presence of the next great thing - the Coonley School - http://www.coonley.cps.k12.il.us/.
Through the intercession of Rahn Emmanuel and Alderman Gene Schulter this school has received a massive influx of financing and the support of a savvy and involved population of parents and kids - http://friendsofcoonley.org/2007/09/what-is-friends-of-coonl
- to become a regional gifted school. Of course you have to be within the oddly shaped boundary of whatever school you wish to attend. Aside from residing within the boundaries of a school you have the option to test in to schools such as Decatur - but then you will be subject to the vicissitudes of the curve the year your child takes the test. Plus you will have to shlep your child to whatever school outside of your local neighborhood that you've been accepted to.
At the end of the day the choice, of course, will be yours. My wife and I, with two young children, are committed to remaining in the city. We will seek the best possible public school but hold private schools as an option. I have clients with children who I have represented in buying homes in River Forest, Oak Park, Hinsdale, and the North Shore. I also have clients in numerous Chicago neighborhoods with children in public schools as well as in private programs that include Parker, Latin, the British and French Schools, and denominational schools. With each, as with you, it will be a personal and financial decision.
With respect to your original question, if the home you are considering is within the Coonley district you are good to go with your Lincoln Square home. Outside of Coonley you will have to do more legwork for a public education but it is still doable. Baseline reality, though, is that living in the city with kids in school is a viable option.
Best of luck with your decision making and house hunting.
The Real Estate Lounge Chicago with @properties chicago