> effectively everyone lives in a city in the modern world, and while their housing might not be air conditioned most of the major buildings are. In an emergency like this everyone would just crowd into office buildings and shopping malls.

Heat waves spur increases in violent crime, limited in some degree by the fact that people stay in their own homes in really hot (but not uninhabitably so) heat waves.

Heat waves that go further to where the only survivable solution is pulling people out of their homes into large communal shelters (whether they otherwise serve as offices or shopping malls or whatever) is...going to be very ugly.

And that's even before considering whether the facilities or supporting infrastructure (plumbing, electrical systems, etc.) are adequate to support these places being used as shelters. Almost certainly they won't be until it's happened and failed a few times, and even after that the places that most need it will be the least able to finance the necessary upgrades.