What is a processed carbohydrate to you? Only refined wheat flower, white rice, and sucrose? These issues started with the shift to agriculture, and the cultures I believe you're describing are all agricultural. That's why civilization exists at all.

Hunter gatherers ate all sorts of things. But starch and sugar was not easily accessible. Wild rice, for instance, would be painstaking to gather and consume in large amounts. Roots and tubers would be small and have lots of antinutrients, keeping you from eating a lot of them. Fruits were seasonal and generally tiny.

If I were a hunter gatherer, in the absence of fresh hunted meat, I would get more nutritional value out of worms or larvae in the dirt than a handful of tubers that may give me a stomachache. Insects are animal protein (and fat).