Does that matter? Depends on your point of view. What I meant by "how new it all seemed" then tells the tale here. Baruch could maybe see the future in 1945 because of his experience and vision. But by 1970 or thereabouts, almost anyone who touched it could see it, and millions were increasingly able to touch it, precisely because it was was commercially produced and generally available. Still not ubiquitous, as they are today, computers were nevertheless showing up in every corner of life.
So it's not that I think computers or information as a processing as a concept, or even as an occasional reality, were new around 1970. But computers as a phenomenon that would define the way interact with each other and the world, and computer programming as a skill that anyone could acquire, really were.
But Turing complete computers predated that. In fact Turing's own design for a Turing machine was NOT a stored program computer. And you're right that the modern idea of programming postdated stored program computers.
But computers are older. As you pointed out, arguably thousands of years older.
However I maintain that automatic tabulating machines were on the path to modern computers. Early accounting applications were based on them, as were key parts of the technology used. Like punch cards.