Ice has a lot of thermal capacity. It’s not so much that it’s significantly colder than the drink, but that its energy-absorbing capacity is locked away as “latent heat of fusion”. When you freeze water, it drops quickly to zero C, then hovers there for a while as the water changes to ice. Toi a casual observer, it looks like nothing is happening. When it melts, it takes the same amount of energy to change it back to water, which comes from the drink you want to cool.
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