They actively use this design in large buildings (with a modern twist). Its known as a chilled water system: https://hvactrainingshop.com/how-a-chilled-water-system-works/

Or you have ones that do not run at all during the day, and only chill/freeze the water at night on excess power/cheap power: https://www.buildinggreen.com/news-article/making-ice-night-cool-buildings

The second system I linked would then let the ice slow melt over the day as its way of actively chilling air passing through its exchanger.

These systems work by chilling water instead of air, which has a much higher heat capacity. Meaning, it can accept much more thermal energy per unit mass before raising its temperature by 1 kelvin. You are able to build a single, very well designed, and efficient refridgeration unit that can provide HVAC services to up to multiple high rise buildings. This reduces waste and reduces the usage of coolant/refridgerant.

This system can be reversed in the winter (heating the water instead of chilling) with geothermal heat, solar heat, or if no "green" options are readily available, natural gas direct fire heat can be extremely efficient compared to electric coil