I'm not missing it at all. I believe (and many, many studies back it up) that the benefits are small. 8 home games/year. Maybe some concerts? I think the economic benefit from 10K new residents greatly outweighs the benefit from 8 home games where most fans drive in for the game and then drive home after. And like I said, there are many studies that back it up.
I grew up in AH so I'm very familiar with the area. The town isn't growing because there literally isn't anywhere left to build there. If new housing were built, people would come.
Again, where would those people come/move from? Using that area for residential development will be pretty useless. For residential development to make sense, be justified, and be net positive economically and from tax revenue purposes, you need one or both of: 1) population growth in the area, and 2) business growth. You would also generally need housing shortages. None of these factors exist.
That town and the surrounding area are not growing because there is no population growth in the area, or in the Chicago metro in general--namely, in the Illinois parts. The Chicago area (particularly the parts in Illinois) has not seen any measurable population increase almost 20 years.
As for the business side, what would those people be moving for? Existing businesses in that area, at least ones that are big enough to result in substantial population growth (e.g., Motorola, which was there at one point) are gone and not coming back. Are you aware of any businesses moving or expanding there?
That area has NEITHER. Most people living there work in the city (unless they are working in school systems, health care, or services/entertainment).
So, as I mentioned, those would be people moving within Cook county, or at most from one of the other counties in Chicago area.
As for your last point, about studies showing that the benefits are small, I'll admit that I have not done the research there, but I'm going to guess that this mostly pertains to situations where the taxpayers were asked to pay for the stadium. This is NOT the case.