I would take anything in the FREIDA database with a grain of salt - the data is put in there by the programs themselves and there's not necessarily any consistent definitions. Use FREIDA for geography, then go to the program website.
For trainees, the most important thing to recognize is that good training exists in many different places and in many different settings. What matters is what is fits best for you and your future career goals. For example, being someone who learns best by reading and then supplementing with a little bit of patient load is going to struggle at the children's hospital where interns are expected to carry 12+ patients and learn all about RSV or CF or DKA because you've taken care of 100 patients with those conditions. Likewise those who learn best by doing are not going to get the fullest training if they enter a program where the patient load is smaller and your attendings expect you to know how things present in textbook ways.
Freestanding children's hospitals are most notable in the fact that ancillary staff and services are exclusively pediatric focused. As you may have already seen in medical school, pediatrics is rarely at the forefront of a hospital's focus and often gets relegated to second class status. From RT's to EMR's to equipment in the emergency room, children's hospitals that exist within larger hospitals may not have all the bells and whistles you'll find elsewhere. This does not mean that training or care is subpar, just that it's different.
From a training standpoint, free standing children's hospitals typically have more pediatric subspecialty coverage, but not always. Thus it's always worth it to explore and ask on interviews what's there. And location matters. A community program in Chicago is likely going to have a lot more subspecialty exposure than one in rural Texas. I agree that if you are considering fellowship, there is a lot to be gained by training at someplace where you can have consistent exposure to the subspecialties you're interested in. I think there is also a benefit to a program having a fellowship in the field you're interested in, not so much due to the ability to match there later, but in getting to see what fellows do and how they function. That can be a valuable resource and set of role models. Again, it comes down to fit.
As for the Chicago programs: there is a pretty defined set of roles filled by the children's hospitals in the city, and it's more complex than simply university vs community based.
Northwestern and University of Chicago fit your more typical free standing children's hospital model, both with world class subspecialty care and unique programs that cater to patient populations that you don't see on the peds board exam. Both institutions prioritize research and have fellowships in most, if not all, available board certified pediatric subspecialties.
Rush, academic but not freestanding, but fits a middle ground and comfortable spot with a focus on excellent clinical care, broad subspecialty coverage but not the same sort of unique programs, and fellowship programs in a handful of subspecialties. Research does occur among the faculty, but the
Loyola is academic, but not freestanding, has had a period of turnover in the faculty and has a more limited assortment of subspecialty coverage.
The Advocate Hospitals are within the same system but have 2 different residencies. People get them confused all the time, but I think Lutheran is the free standing children's hospital in the SW part of the city, which has more subspecialty coverage.
University of Illinois-Chicago is academic but has a strong devotion to care for the underserved. The subspecialty coverage is going through a transition because of a change in the department chair, and the problems associated with the State Budget which is a political mess at the moment. There is some subspecialty coverage, but it's limited, and research is not prioritized by the faculty at the moment.
Stroger is at the famed Cook County so there is a strong conviction for the underserved, but the program itself also takes a lot of IMG's so there is an interesting global health lean there as well. There is a unique coverage set up between Stroger and Rush in which some divisions are separate, others cross cover for each other and others are combined. So there's actually more subspecialty coverage here than at UIC.
Mount Sinai is something I didn't know existed until another poster brought it up here on SDN. Their website is more focused on community health and as far as I can tell has limited subspecialty coverage.