If the phrase was first mentioned (or invented) by J.J Rousseau way before she was even in France, how come this particular phrase is so strongly thought of as being spoken by her? The conotations are, of course, clear and since there was no shortage of sources for twisting facts into bad PR for her, I can very much imagine why it happened. However, I'm wondering if we have any reasonable clue as to what was the journey of that phrase into her historical mouth.
Rousseau does not attribute this phrase to anyone in particular, except a "noble princess". There has been several suspects as to whom he could have been making a reference, Madame Victoire, Madame de Pompadour or Madame du Barry. In any case, he was not at court and this phrase was not found anywhere else at the time that could have lead him to make assumptions. So it is purely anecdotical and certainly invented from scratch (even though similar says exist elsewhere).
Famous German writer Erich Kästner attributes the let them eat cake to Marie Antoinette in his book 1931 Anna Louise and Anton (Pünktchen und Anton). And Kästner was and still is a famous writer, popular with kids ( he wrote a lot of children's books). Most likely he knew that Marie Antoinette never said that (some would say that it would be contrary to her personality) but attributed to her anyway for the sake of his story.