Although there are birds in southwestern Africa, the general carrying capacity of the land west of the 380mm isohyet is such that the overall biomass might meet your conditions. The result is that Khoesan hunters don't seem to have employed fletchings; these arrows at the Peabody bear out the mode of 'bushman' design (who knows what specific group--that description isn't helpful). The sinew wrapping at the ends presumably aided the durability and 'trueness' of the arrow, but that's a question for an anthropologist or devoted archery scholar. Based on the Grayson catalogue (see below link) another possibility is that sinew aided the use of substances that might be applied to the arrow (or coated barbs), assuring its delivery to the target (see entry 141, p.139). A few examples have pitch over sinew.

A lot of African arrows seem to have omitted fletching altogether, but people in other environments, sometimes also low in biomass relative to other areas, did develop fletching. The baTwa in the Congo Basin (or pygmies in colonial parlance) are a good example. Although we can't say 100% that the presence or design of the fletching isn't a hybrid of American or Eurasian methods learned after 1500, in the nineteenth century travelers recorded that dried leaves formed the material, and sharpened palm-leaf stems formed the shafts, as you see in this instance on page 143 (entry 144--PDF warning) and can find in many auction collections online. That document in general has a lot of well informed 'traditional archery' commentary, as it comes from the notable Clarles E. Grayson Archery Collection at the University of Missouri - Columbia museums. Some of it's surprising; I had no idea that Aka crossbows were a thing, myself. Some areas did see feather fletching before the twentieth century, but there's a lot of variability. The whole thing may be of interest to you.