While it's true that you won't find published Carbonara recipes pre dating 1952, the Lazio region has had for centuries pasta dishes based on the same ingredients. And they are thoroughly documented.

Both gricia and amatriciana, too other famous pasta dishes from the same region use the same cheese (pecorino) and guanciale. In fact carbonara is nothing more than a gricia with egg yolks.

It just makes no sense to have parmiggiano or french cheese in a recipe coming from a region that did not have these ingredients in the first place and are not part of its culinary history.

And thus the point of authenticity is into rooting where the recipe originated with local ingredients.

Anybody's free to change the recipe all they want, but to call it carbonara when ingredients don't match is misleading the customer expecting a roman dish with roman ingredients.

The point of a word is to convey meaning with a short string of sounds. The meaning of a word referring to a dish would normally describe the taste of the dish and what it's made of in general terms, because as you've pointed out, recipes are subject to individual variation. To say that a restaurant should not serve a dish called "carbonara" made with French cheese to me sounds similar to saying that an Italian carpenter should not sell a mahogany table and call it "tavolo in legno", because they don't grow mahogany in Italy. Who cares where the ingredients come from if the dish tastes good?

But pasta alla gricia only gets you back to the 1920s; I think the one Roman pasta that goes back centuries (and is in the same clade as carbonara) is cacio e pepe.

Ironically Italians criticize Americans for non-authentic "Alfredo sauce" but that has its origins in Rome in roughly the same early 20th century time period.