Some of that described my daughter’s criteria four years ago - and I will comment as far as they fit Barnard, since you had tagged it:
Barnard has great programs and professors for social sciences, English and arts. In fact these are my daughters majors and minors, and she very much enjoyed the lecturers, enormous breadth of courses, and variety of topics that Barnard, and the large university catalogue offered. Those are also the disciplines where she picked up internships in the city.
However, at Columbia U you’ll need to “temper” expectations as far as the classic “school spirit” for the football team that she might know from high school. It won’t have nearly the same vibe. Obviously Columbia U competes in the Ivy League, and in certain sports (such as fencing) is a powerhouse, and Barnrad women are on several competitive female teams in various disciplines. However, the stadium is a (free shuttle) bus ride 3 neighborhoods away, and you won’t see 33,000 university students wearing gear and creating huge excitement on campus on game days.
As far as theatre/music - in general, this too will be different at colleges vs. high school. While in tax-paid high schools, everyone is pretty much welcome in any club, in colleges there usually is an interview/audition process for many such extra-curricular activities. This is not specific to Barnard.
Having said that, participating in two performing and visual arts related clubs was a big factor for my daughter’s happiness, creating a counterbalance/outlet for academics. So yes, a big university like Columbia absolutely has many extra-curricular opportunities for students interested in arts, it might just not necessarily be in the exact manner that your daughter is currently thinking of.
The first two are a good description of Barnard - it comes with all the benefits, opportunities and activities of a large, co-ed university in the “greatest city in the world” with countless cultural offerings of all kinds at all times, while also having a small little campus to itself when the hustle and bustle gets a bit much.
Much of the time will be spent on the university campus, which is actually quite nice, going to classes, using one of the libraries, any of the dining halls, going to the gym, etc.
Now - as far as warmer than Illinois… I suppose technically that’s true for New York, based on statistical data.
My daughter had stalled deciding for Barnard until the last possible moment - and now she’s actually genuinely sad that soon she won’t be passing through that gate anymore. It truly does feel home to her (even though, like very many upperclass-women, she’s been living in apartments in neighboring blocks.)
However… for hospitality programs, there might be schools that are much-better known for those studies!