Why is coffee mostly hot brewed? Why is coffee always made with hot water ?

I know there is iced coffee, but even in this case, coffee must be made with boiled water before.

My simple coffee machine heats water into vapor, then rises it to the top of the machine to fill the filter and turns to coffee. It's obvious that it's impossible to do this with cold water and a traditional machine. But if it was, would the taste be the same as of a cooled traditional coffee ?
Let's take a look at what turns coffee grounds and water into coffee:

It's an extraction process where aromatic compounds and oils and caffeine are extracted from the coffee bean into the water. It's a "standard chemical process", done in millions of kitchens and cafés all over the world.

Basic chemistry will tell you that the temperature of the solvent and other environmental parameters like pressure typically determine how fast this happens, but also the ratios of extracted substances, if applicable. Surface area comes into play as well, that's why e.g. coffee for espresso machines is ground finer than for filter machines. Using the same batch of beans in an espresso machine and a drip coffee maker gives you two totally different taste profiles.

The same applies for extraction processes that use cold water, a method termed "cold brew" and discussed on this site as well. By the same rules as explained before, this means you need significantly longer extraction times (we are talking hours here) and should expect a different taste.