As someone who dabbles in archery, I find the idea that a person would invent effective bows and arrows that far back, in deep prehistory, to be... well, dubious at best.
The arrow is a wood-and-cord spring for firing small bladed missiles. It took prehistoric humanity a good long while before they realised they can attach a sharpened stone to a straighter branch or to a short handle and use these contraptions as spear, javelins and early axes. The first axes - handaxes - are called that because they didn't actually have any sort of handle.
Why am I saying all of this ? Because it took a good while before humanity invented the first javelin or throwing spear for more effective hunting of animals from afar. As much as that improved human chances at successful hunting, it was still quite an ineffective tool for catching prey. Fast-forward to thousands of years later and you finally see the slow emergence of the spear thrower. Such a simple concept, they should have thought of it immediately after inventing the hunting javelin, right ? Well, no. We might think so with the benefit of hindsight, but in a species with low population numbers and a focus on day-to-day survival, would researching tool and weapon improvements really be the first thing crossing everyone's mind ? Hardly.
But back to the spear thrower. Despite its almost ridiculous simplicity (just another piece of material used as a lever for throwing the spear), this is the first "wooden spring" type of weapon, the first man-amplifier, ever. The bow was just a more complex (and quite a bit later developed) iteration of the same concept. Hindsight can be incredibly misleading: "But a bow is such an easy thing, why didn't they think of it much, much earlier, right after they had thought up the spear thrower ?" Well, would it occur to you to take a piece of tying material (remember, there are barely any threads made of plant material before the Neolithic, just ones made from animal material) and tie it in a careful and very deliberate way to a very deliberately carved and smoothed wooden selfbow ? If even inventing the spear thrower took a good long while, how long do you think the invention of the bow will take ? Quite a bit, I'm afraid.
Looking at even much more complex technology, this isn't absurd at all. In the last 500 years, we've repeatedly re-invented designs for repeater firearms or for steam-powered wagons usable as commuter vehicles, but they never really took off in their first iterations. To get an invention right, one must make a lot of trial and error attempts first, and keep at it on a long-term basis, all the while not being discouraged. Also, do you think most tribesmen of the pre-Neolithic would be thrilled if some dreamer in their tribe wasted precious time by fiddling with (and wasting) materials on some sort of daft "bow" idea ? I doubt they'd be that tolerant, especially if they had the impression that he's not getting anywhere and is staying at home in the process, lowering their chances at successful hunting.
Like with everything in human prehistory, every single breakthrough and highly influential invention/discovery took quite a bit of time to get perfected and become accepted in wide use. Theoretically, maybe some fellow invented an awesome Stone Age bow and arrow as far back as 50 000 BC, but if his culture refused to adopt it and use it, and he "took the secret to his grave", then the innovation of the bow was snuffed out early until some other guy or gal in that same or completely different culture thought of it again, as a scientific/tool concept.
I therefore doubt you'll be seeing even pre-Cromagnon and pre-Neanderthal representatives of Homo leaving Africa with bows and arrows, nevermind pre-Homo hominids doing the same. The simplest of wooden bows might look easy to make at face value, to our jaded contemporary eyes, but how many people from over a hundred years ago thought of inventing something as mundane and deceptively simple as the clothes' zipper ? Very few, and it took a while until the thing was patented and started spreading as a legit improvement for modern clothes.
It's similar to how in antiquity, you couldn't really get a society with philosophers and scholars unless that society was already wealthy enough, had well-established trade and a surplus of food to live off of. This allowed certain parts of society to devote some of their suddenly available free time to things like thinking, speculating, or even more actively pursuing science, experimentation, inventions, or arts.
Concerning "the bow being so easy to make, everyone must have surely come up with it !" claims: Some cultures that had been isolated for a very long time never even bothered to invent bows on their own. Look at most of the Australian native cultures. They were skillful, smart, etc., but aside from a few tribes in the north getting the bow via contact with overseas native cultures, most of the natives never created bows. They were perfectly happy with their woomera-s (and other names used for their spear throwers) and kept using the thrower+spear combo for hunting for tens of thousands of years. And they weren't any worse off for it.
