Ah, well, this is actually a much deeper question than it may seem on the surface. However, unfortunately there's a huge chunk of the story which has received almost no research or scholarly attention.

So, let's start at the beginning. The origin of the @ symbol dates back to the middle ages. At that time it was still common for many trade goods to be transported in "amphorae": large, thin necked, stoppered, two handled ceramic vessels, as a hold over from roman times (eventually these were more or less entirely superseded by wooden barrels). These were used to transport things like wine, olives, oil, and so forth, and had been in use in Europe from the neolithic. It also became a common weight/measure (much like "barrel of oil" or "tun of wine") independent of which container was actually used to transport the item. Anyway, amphorae is a fairly long word, and at one point someone had the bright idea to abbreviate it with a little 'a' with a swirly circle around it, the proto-@ symbol. This history echoes in the names for the symbol in several romance languages, particularly Italian where "anfora" is one of its names.

This is how the symbol came into being, but for many years it was not necessarily very commonly used. Nevertheless, it did gain expanded uses. There were various instances of it being used as a shorthand for one or another word that started with "a", but the main use became as a convenient abbreviation for documenting rates or prices for things in business. For example, "3 hours @ $10.75/hr" or "2.75 lbs @ $0.25/lb", that sort of thing. That usage is why the symbol ended up on typewriter keyboards and later on computer keyboards (or, more properly, teletype machines) and was around for being turned into what it has become today.

But wait, there's more. And here is where this gets a lot more murky because there has been incredibly little research. Through the 2nd half of the 20th century, at least, and probably longer, there was a sort of basic skillset related to "office literacy" or perhaps just "clerical literacy". For example, it was common for a lot of people who worked in an office to understand the principles of filing, which is a rapidly obsolescing skill these days. There are lots of little things that would have been part of this common skillset. For example, the way to conduct a "formal" meeting. Having someone designated to take "minutes", reading the minutes from the last meeting and discussing any unfinished old business, bringing up new business, etc. A generalized agenda format. This would have seen common practice in many business and office settings as well as in various clubs and organizations. This sort of thing has become a lot less ubiquitous in business today substantially due to the dominance of the more ad hoc and less formalized culture of tech, which has swallowed the world.

One part of this basic business literacy skillset was abbreviations. Things like "ea." for each, "w/" for with, "w/o" for without, the "@" symbol, and other more specific abbreviations like "net 30" or "YTD" or "Q1". In an era of hand written and hand typed messages, these forms of shorthand were quite common. And people with this skillset would have used those abbreviations in many contexts, perhaps spreading some level of partial "literacy" in these abbreviations out to others they interacted with. For example, teachers tended to have this skillset in order to be able to work within the bureaucracy of the school system, and it would be easy for them to spread the use of it to their students.

All of which is to say that the use of the @ symbol was a bit more prevalent than within the solitary "@ rate" business use outlined above. But exactly how prevalent is something that nobody's done the research on.

So, to get back to the specific question. Would the average person (here I'll substitute American) from the mid-20th century be expected to recognize the symbol? Honestly, I really don't know, and I don't think anyone knows for sure. A great many people would have known it. And a majority of college educated people or office workers likely would have known it, but that's not necessarily a majority of the entire population of the US.