Where cold is the norm
Mugicha and boricha live in a fridge through a Japanese and Korean summer. Oksusu-cha the same. Agua de Jamaica is sold from glass jars in Mexican markets. Karkade is iced in Egypt in summer and hot in winter from the same material. Tereré is the cold form of mate and is more common than the hot one where it is drunk.
Two ways to get there
Brew hot and chill, or brew cold from the start. Hot-then-chilled is faster and extracts more; cold-brewed takes hours and gives a cleaner, less bitter, less astringent drink. For anything with tannin or bitterness to manage, cold brewing is the better result.
What chilling does to flavour
Cold suppresses sweetness perception and sharpens acidity, so a drink balanced hot will taste sourer cold. Every tradition that serves these iced sweetens them more heavily than the hot version, which is a calibration rather than a sweet tooth.
Colour is less stable than you think
Beetroot browns as it stands. Butterfly pea shifts the moment anything acidic reaches it, so a citrus garnish changes the colour rather than just the taste. Both are worth knowing before making a jug for other people.