There are plenty of cookbooks and literary reference to back up the idea that omelettes -- by that name or something recognizably close -- were well into the French and English mainstream by the 16th C. I tried researching contemporaneous recipes but after spending as much time as I cared to spend came up with zip.
I know when Mere Poulard started serving the famous Mont Saint Michel omelettes -- around 1875 -- but it was my impression her omelettes were simply a variation on a common, Norman style and had been around the area since Moses wore shorts.
In any case, I stand by my contention that plenty of "authentic" omelettes in France show quite a bit of brown; and that the omelettes of Mere Poulard are among them; and that they are "classic" by any reasonable -- other than purely technical -- sense of the word. If by "classic" we mean haute cuisine of the early twentieth Century, as propagated and described by Escoffier, and refer specifically to the sort of omelette aux fines herbes Escoffier served, then "no brown." If by "classic," we mean typical and widely and long established, then some brown at least is okay.
I happen to like omelets (not the spelling change as we segue to the US), just about any which way except overcooked. I don't fancy myself a great omelet technician, but have no trouble getting some brown on the eggs without toughening. Nor do I have any trouble cooking the eggs to the French semi/custard consistency without browning.
I don't think I've ever seen a "classic" (in the hyper-technical) French omelette which has been flipped. The whole idea of that sort of omelette is that one side of the egg mass is completely but barely cooked, while the other side is not quite or barely just set; so that when the omelette is rolled, the consistency is indefinite -- neither one thing nor the other, and never dry. If you want well-set eggs, you're barking up the wrong tree, at least as far as the Escoffier "classic" -- and definitely no flipping.
That doesn't mean I look down my elegant aquiline nose (okay, it's neither, not to mention broken a few times) at flipped omelets. One of my most favorite omelets is "salami and eggs pancake (not folded) style." I'm not about to turn that nose up at a tortilla Espanola either.