One of the claimed benefits of cold-water extraction, in addition to convenience, is the lack of acidity, so those who have sensitive stomachs can still enjoy coffee. Since one's stomach produces hydrochloric acid for digestion, and the stomach naturally is an extremely acid environment (pH under 2), and coffee pH averages 5 to 6, I'm not sure coffee is the culprit. But I'll leave that to the gastro-intestinal doctors.
At least one writer claims it's a myth that coffee causes an acid stomach. More important to me is that acidity is a required part of the structure of taste. I distinguish between palate acidity and stomach acidity so I can focus on taste, leaving the stomach part to the experts. The presence of some acidity in the taste greatly enhances the appreciation of other flavors. This is especially true with some sweet things. Fruit jam, for example, may just coat your taste buds with cloying, fruity sweetness. Smart jam-makers add some lemon juice for acidity. Then your mouth lights up: the cloying quality diminishes, and the fruit flavors become more prominent.
It's the same with wines. I notice it particularly with white wines. If there is some acidity, we will enjoy the floral qualities of Sauvignon Blanc or of Riesling much more than if the acidity is missing. When the acidity is missing, most people taste sweetness--even if in fact the wine is not sweet but floral. When the acidity is present, the wine is in better balance and your palate is enlivened.
But I digress. My point is that I want some acidity in my coffee to help me perceive all the flavors. I find cold-water coffee to be lacking in acidity and therefore in flavor. Perhaps the enthusiasts of cold-water coffee are using sugar or milk that might mask the lack of flavor.
It's well known that the temperature of the water is crucial to good brewing. The lack of temperature is what causes most coffee made in inexpensive automatic drip makers to be either insipid, from too low a brewing temperature, or bitter, from too long in the brew chamber.
Brewing is the method we use to extract the good soluble solids from coffee. Not enough extraction leaves an insipid brew, as in cold-water extraction; too much extraction from brewing that is too long or too hot produces bitterness. But brewing the correct grind for the maker at the right temperature (195 to 205 Fahrenheit) and the right time (3 to 5 minutes) gives you what we have all been seeking these last 1,000 years: a good cup of coffee.
For a good cup of coffee--hot or cold--brew it hot for flavor, then serve it at any temperature you want.