However, https://en.wikipedia.org/wiki/Ice_house_(building) talks about Zimri-Lim's famous icehouse, but also says:
In Texas, former ice houses are a cultural tradition. Ice merchants diversified to sell groceries and cold beer, serving as early convenience stores and local gathering places. The widespread 7-Eleven chain of convenience stores in the U.S., first known as Tote'm Stores, developed from ice houses operated by the Southland ice manufacturing company in Dallas and San Antonio in the 1930s.^[12] Many Texas ice houses have since converted into open-air bars. In central Texas, southeast Texas (especially the Houston area), and the Texas Hill Country, the word "icehouse" has become a colloquialism for an establishment that derives the majority of its income from the sale of cold beer.^[13]
Laura Ingalls Wilder's novel Farmer Boy https://archive.org/details/farmerboy00wild_0 decribes her husband's New England childhood; one chapter is a story of how he and his father filled their family icehouse with ice bought in the winter from the Frenchmen who cut up the river ice. His father was a prosperous farmer, one of the richer men in town, with several horses and many cows and pigs, but he was no king or governor; his son got up at 5 AM every morning to milk the cows. While Farmer Boy, like her other books, is fictionalized, she did try to remain broadly factual in what she described; and, more generally, she is fairly reliable about what kinds of things people of her time considered normal.
The late modern history of the ice trade is really quite remarkable, and it's unfortunate that the article gives it such short shrift. Starting in the 01700s, ships carried ice from Scandinavia and the Great Lakes all the way around the world, until the invention of practical heat pumps (implicit in Carnot's 01824 heat engines) in the late 01800s made it cheaper to produce ice locally anywhere on Earth's surface with a steam-engine. The key to this was "chimogene", a fraction distilled from petroleum that boils at or below 0°; now we call it propane and butane. But it wasn't until after 01900 that the last ice ship sailed, and daily ice delivery for home iceboxes was common in the United States until the 01950s. The Amish still practice it today.
Even today, in the United States, air conditioning systems are rated in "tons"; a "five-ton air-conditioner" provides as much cooling as a daily delivery of five tons of ice.
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The fundamental physical phenomenon that makes icehouses possible is Galileo's square-cube law. A one-meter cube of ice weighs about a tonne and has six square meters of surface area through which heat can enter to melt it, 6 m²/tonne. A ten-meter cube of ice weighs a thousand tonnes but has only 600 square meters of surface area, 0.6 m²/tonne.
So, with a little straw around it, it can remain frozen throughout spring, summer, and fall. Water's humongous enthalpy of fusion of 333.55 kJ/kg is 333.55 MJ/tonne or about 300 MJ/m³, so such an icehouse can soak up 300 GJ from its environment before it loses all its wares. For a seasonal thermal store, that's about 10–15 kilowatts, depending on how long winter is. Dividing by dry straw's thermal conductivity of about 0.05 W/m/K, we get 190 kilokelvin meters; divided by 600 m², this gives us a maximum thermal gradient of 300 K/m.
So, if the outside of the icehouse averages 30°, 100 mm of straw is enough.
Nowadays many building air conditioning units use the same principle on a shorter timescale, freezing water when electricity is cheap and using the ice to cool the building all day long. And similar principles apply to other phase-change energy storage systems, whether using sal mirabilis, its eutectic with halite, or the aqueous eutectic of sylvite; though I expect thermochemical energy storage using muriate of lime and perhaps magnesia to make most of these obsolete, in part because they need no such insulation.