I don't think the author was trying to say that these spices are harvested in the fall and enjoyed in the winter. In fact, I think they are clearly saying the opposite.

Just as cranberries’ fall harvest makes them a natural choice for Thanksgiving, I thought that perhaps the seasonality of spice harvest had something to do with their use during the winter months. However, this doesn’t appear to be the case.

And

Take ginger...The plants can be harvested at any time of year if they are mature and haven’t been exposed to cold or wind.

Or, if they are trying to make that point, they are doing so in a way that includes contradictory details.