Hi, OP here.
This little theory of mine is very similar to that of what John Greene spoke out about 99 cent cheeseburger on youtube video "Crash Course World History 1: Agricultural revolution."
https://www.youtube.com/watch?v=Yocja_N5s1I

I do recommend you watch it.
The point is, this kind of product which requires interconnected production and supply networks (and especially it being widely available to general public, not just the elite) is only available in a system that has achieved societal cooperation on a level which we could call "civilization".

Now, that does not mean it would be an equal or democratic or even fair society, even ancient Romans (which were slave-owning horrendeous bastards, not going to lie) enjoyed something aking to a hamburger and ice-cream. They might not lived in a technologically very advanced civilization, but IT WAS a civilization.
(there are whole history books written about ice and milk being transported from the pastoral mountain regions into low-land cities, which led to accidental discovery of ...well, something aking to ice-cream).

You don't necessarely need even electrical knowledge for it, but it certainly helps. If you have it, it many steps in this process become easier and the end product more easily available for the general public. And thus, reach of your civilization becomes wider and more available for the masses it contains.