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You originally need someone scientifically clever who can figure out how to keep the ice melting en-route.
We are not better at science today than the people of the past. We have better tools for observation and for data capture. We have better communication tools, to distribute information and to educate. We leverage these tools. That leverage is a large multiplier. The base cleverness of the people is the same.
Those men who were involved in the movement of goods over a distance, they knew methods to manage the temperature of the goods, just as they had to manage the temperature of their bodies over those distances. They were interested in efficiency, profit, and their own leisure time, the same as people of today. They transported ice and foods without refrigeration machines and the phrase "scientifically clever" describes some of them well.
+1 regarding what SixOfSpades and mightymushroom posted about the methods.
I don't think most people would want lake ice in their tea, wouldn't it be full of dead bugs and silt? Gathering snow and compacting it sounds like it would produce a product people would want to consume.
I posted this earlier, "People would be sent to do the work of creating or improving pools, so ice would be available in larger quantities and higher quality...". You should be able to imagine the type of masonry activities involved in this. Those masonry activities are the same ones used to create water reservoirs today and have been used for thousands of years.
The product being delivered over a distance is the "potential thermal energy", Snow and ice are both frozen water, but ice holds more of this "potential thermal energy" per cubic meter. This means it is easier to transport the same weight of frozen water to the destination.
If you put a metal bowl in the freezer with 2 liters of water in it, give it a few days to freeze completely, bring it into a cool room (subterranean rooms would be used in the past), apply heat to the bowl so the ice can fall out of the bowl, then use a metal tool to chip at and shave the ice. Once your technique improves, you will have ice shavings that you can use to create a cold drink that would make any over-heated person happy. If that person did not have refrigeration, it would be a marvel to them. A noble would use this luxury for their court and guests, to impress them, just as a merchant would sell this luxury to those who wanted to impress others.