The problem of a lack of free public indoor spaces isn’t limited to New York City; I live in suburban Connecticut, and (barring public libraries, and the occasional bus shelter or other wall-less structure on public land) indoor spaces accessible to the public are either retail spaces or member-supported ‘clubs’ with limited access (often, non-members are allowed in only by invitation or are only allowed during particular time periods).
I get the impression that in the past (say, prior to 1950) public spaces were more common and more vital, with community organizations like clubs, churches, and lodges being less closed-off simply because socialization and other forms of entertainment for those without any money was necessary. Salon parties & coffee houses also catered to this use case, and it seems like coffee houses would cultivate social groups and bring in events like scientific demonstrations & political speeches during the nineteenth century, as a means of drawing a crowd some of whom would purchase coffee, the same way that bars will occasionally bring in gratis musical acts. By the 1960s, at least from the perspective of europeans, these spaces were already gone or commercialized: the concept of a shopping mall was created specifically to produce european-style public spaces in the united states, with shops and commerce being in the original conception an unimportant side-effect & features like benches, squares, and food courts being more central. (Of course, shopping malls essentially suffered through a very long bubble, as a result of real estate loopholes; they were everywhere and extremely commercialized up through the early 90s, at which point they started going out of business.)
The best idea I can think of for bringing back public spaces is to expand & publicize the low-key social spaces that already exist in or associated with libraries: libraries are already indoors, and they already essentially deal with people, but most social spaces in libraries in my experience are exclusive & by reservation only; with good sound-proofing & clever layout, social spaces & quiet spaces within a library can be segregated and the kind of free-wheeling nineteenth-century-coffee-house experience can be reconstructed to some extent.