Others with much more expertise than I have written about cold storage, and storage of ice, before home-use refrigeration.

/u/valmyr5 on ice in antiquity.

/u/bentresh on sumerian ice houses.

/u/j-force a little later in time.

/u/ashkenazeeyankee and others on ice in the US before home fridges.

It won't surprise you that ice cream follows ice. I think your distribution model is a little off, though: the ice cream parlor and the soda fountain were where people would get frozen or cold confections more than at home, for just the reasons you've intuited.

For further reading:

Laura Weiss - Ice Cream: A Global History;

Jeri Quinzio - Of Sugar and Snow: A History of Ice Cream Making.

You don't need electricity or a heat pump driven refrigerator to make or store ice cream, you just need cold.

Firstly, I will mention that in many inhabited parts of the world conditions at or below freezing can be had for weeks or months at a time for free just by taking advantage of the wonderous invention called "the outside". Indeed, it's important not to discount this resource. Many people who live in higher latitude regions will be able to regale you with stories of the utility of ambient conditions as a ready means of food preservation during the winter months. However, it has also been extremely valuable as a resource for producing secondary foods. For example, if you take ordinary naturally fermented apple cider (which might have an ABV of a few percent) and you freeze it while removing the ice you can effectively increase the alcohol content through freeze distillation, producing a high ABV beverage (up to 80 proof) known as "applejack". And in the process using vastly less energy and resources (no plumbing no heating, etc.) than with ordinary distillation. You can use a similar process with maple syrup to produce "maple sugar", which has been produced in North America by indigenous peoples since before contact with Europeans.

But, of course, you can also take advantage of the fact that you can actually transport the cold from "outside" to other locations where it is more convenient to use. For centuries there has been ice production (via cutting it from lakes or glaciers) and transportation to populated areas, sometimes over very long distances. Ice houses as means of storing potentially perishable foodstuffs and other items have long predated industry. In the late 19th century it was very common for urban dairies to be co-located or nearly so by ice houses so they could rapidly cool down fresh milk and deliver it across the neighborhood daily, where it would be placed in a small insulated box near the doorstep and then transferred into an "icebox" in the home, itself refreshed regularly with fresh blocks of ice. I should note here that the icebox is similar in concept but much different in design from a modern cooler, they were designed for being cooled by mainly one sizeable block of ice and the cooled items were kept on shelves and not immersed.

By the mid 19th century iceboxes and residential ice delivery were becoming fairly commonplace. Around that same time home ice cream makers also came on the market. Frozen treats had long been a component of upper class dining for centuries but were more often a rarity until about the 17th century (one might speculate on the local availability of ice and the relationship to the Little Ice Age which was occurring at the time, though perhaps it was entirely coincidental). Through the 17th and 18th centuries there was an increasing popularity of making sorbets and ice creams, which was substantially aided by the discovery and especially the dissemination of knowledge of the "endothermic effect". By adding salt to ice you encourage the ice to melt, but it takes a lot of heat to melt ice, and this heat is absorbed from the nearby environment, causing a rapid chilling effect vastly more potent than simply placing a dish of cream in a bowl of icy water. The speed of freezing also allows for the creation of ice creams and sorbets with much softer textures than they would have otherwise since they can be continually stirred or churned as they freeze. Recipes for producing ice cream in this way were published from the early 18th century and it became a common offering in many large European and American cities.

The advent of the ice trade helped make home ice cream making feasible outside of the seasons and regions where ice was plentiful. Which gave rise to the invention of the home ice cream maker: a simple device that allowed turning an external crank to keep churning the ingredients while they were being rapidly frozen by a salt and ice mixture. And what do you do with the ice cream after you've made it? Well, you eat it, of course. Keep in mind that in the US in the mid 19th century roughly half of all households had at least 6 people, so even a whole batch of ice cream wouldn't last very long. However, if you happened to have an icebox as well then you could potentially store any of the excess for a while. And that's pretty much the situation as it would apply to individuals. You could get ice cream at a confectionary or a specialty shop, but this was all designed to be eaten on site, not packaged for storing at home. Or you could make your own ice cream at home, but this would generally get consumed promptly.

I should point out that even before electric powered home refrigerators became common, industrial refrigeration did make huge advances even in the late 19th century. Production of ice from refrigeration equipment, rather than being harvested from distant natural sources, started taking over in the ice trade, and similar equipment began being used in the production of ice cream as well. Through the 19th century ice cream production became increasingly industrialized along with all other foods. Instead of being made on site in small batches ice cream was increasingly made in larger quantities in centralized facilities, but it still took until the 1930s before bulk ice cream was sold direct to consumers in grocery stores.