nisht: The Halachic status of an Am Ha’aretz is all D’rabannan. The phrase “Am Ha’aretz D’oraisa” is nonsensical.
nem: That’s not what Rashi means. The Gemara says prevent your children from studying “Higayon”. Rashi explains Higayon as meaning Nach. The reason for this is that, in Rashi’s time, most people who learned Nach were the Catholics. Thus, being too involved in Nach could lead to intermingling with the knowledgeable Catholics, which we tried to avoid. (Fascinatingly enough, many of the Sephardic Rishonim translate “Higayon” there to mean philosophy.)