Although you are entirely correct in a technical sense, and it's common to serve hot brewed coffee cold, the opposite is so rarely desired that it will be considered 'weird' (in the sense of 'unusual') 9 out of 10 times.
Every extra process involved in making a coffee is going to add complexity and time to the workflow, which many cafes will elect to charge extra for.
There is some consternation in Australia about paying more for iced coffees compared to hot ones too:
https://www.broadsheet.com.au/national/food-and-drink/articl...
