Ok, one, you get great schools in most of suburban New Jersey. Most are probably better than IL schools since NJ schools consistently rank higher, almost always top 5 and recently top 1-3 out of 50.
There are various outdoor community pools in suburban NJ and I imagine NY as well. My town alone has 3 separate pool centers (each with multiple pools within them), and they're all clean and updated or in the process of being updated. One of these also has an indoor pool and gym. Membership depends on the town or individual pool center.
There are plenty of community centers in this area. My town has one and I know all neighboring towns do, as well. Many have built in gyms, libraries, spacious rooms, and outdoor areas among other random things like that. In our community center, there are sewing and quilting classses and the PAL cheerleaders often practice inside. The recreational children's basketball teams play there and we recently started an indoor winter artisan market there as well.
You're looking for skating rinks and batting cages? Yes, we have them here. All over.
I'm not sure if you're trolling or just didn't look very hard. You can find everything you're looking for and more here. Sure it may be more expensive but you'll be living in the NYC area (where amenities are plentiful and QOL is great) and that's just how it is. I just googled Park Ridge and there are plenty of NY and NJ NYC suburbs just like it in terms of wealth, schools, and amenities. Places like that are run of the mill around here.
Sorry for the deleted message above, for some reason this message posted twice.