It's only the extreme north that would provide scant carbs--most temperate and tropical areas would have had carbs available much of the year and early humans would have been clever enough to either store excess supplies or steal from animals who did.
While it's true that some food plants have very short seasons others are around much longer or store well. I grew up in Massachusetts and the woods and wild borders there were full of food, so much so that in the fall you could easily store fruit, nuts and tubers for well into the winter. Some berries don't become edible until they've frozen during winter on the bush; although we don't commonly eat them now, many greens that sprout throughout the growing season are edible. And, of course, seeds/pods that remained on the plants could be gathered when green things weren't available.
If you drop our modern "only the most lush, most juicy piece" paradigm and just look for edible, it's all around us.