Major cities also have private libraries where a similar arrangement exists. Public libraries may be too busy and not as stringent in maintaining sound levels. They usually don't have private rooms either. From my experience, they are also not as clean as private libraries.
I've been confused why more people don't take advantage of these "friend of a library" programs or take part in private library memberships. The programs allow non-students to support libraries (although private ones) and they're significantly cheaper than going to cafes or coffee shops. I'm pretty sure most places will want you to buy something everytime to be considered a patron so that you aren't kicked out or pressured for freeloading.
Examples:
* Cooper Union (https://library.cooper.edu/friends/)
* NYU (https://library.nyu.edu/about/general/fob/)
* Princeton (https://fpul.princeton.edu/)
* NY Society Library (https://www.nysoclib.org/)