The view of technological innovation that you're describing in your post is known as diffusionism. For a long time, that view dominated in history and archaeology, especially among those who viewed human history as tale of "great societies" and centers of innovation developing new ideas that either they spread through conquest and migration, or through interaction and exchange of ideas, or both. Politically / socially, diffusionism is especially popular among nationalists and ethnocentrists for this reason.

That's not to say that it doesn't have some value in understanding history. There is evidence that some ideas or technologies did spread via diffusion-- originating in one place or area and spreading outward. The spread of domesticated crops from their regions of origin to other areas is a good example. However, because human history is complex, cultural processes are as well. Diffusion is a lot more complicated, and the agriculture / domestication example bears that out.

While the crops themselves may have moved, the idea of food production arose independently in many regions of the world at different times. The arrival of corn in North America far post-dates its domestication in Mexico and its widespread adoption as a major source of food in that region. Corn diffused to North America via trade and (probably) migration into the American Southwest and from there into Eastern North America.

But the concept of food production did not. Corn / maize arrived in Eastern North America at least 2000 years after local groups had already domesticated a series of indigenous crops, including sunflowers. So while the crop itself diffused, the idea did not. And the crop was selectively bred in different regions based on the desires of local populations, environmental conditions that favored certain strains, and so on. So diffusion isn't enough to explain the process.

Tracking the history of innovation and spread of ideas and technology is tricky because it's not nearly as simple as "this thing was invented in X place and people from there brought it to Y place."

This is why diffusionism as a default explanation-- and especially as the only explanation-- is faulty.

At the same time, multiple instances of independent innovation-- which was the go-to explanation when diffusionism started to fall out of favor-- also doesn't work as a default, because (as explained above) we know that some ideas and technology did originate in one area (or in only a couple areas) and spread rapidly because of their benefit over other similar tech or ideas.

And even if examples of a particular idea or technology did originate "independently" of each other, are they actually independent, or simply such good ideas that they arose out of more widely shared ideas or precursor technologies.

The concept of evolution by natural selection is often attributed to Charles Darwin because of his publication of On the Origin of the Species in 1859, but in fact a colleague of his-- Alfred Wallace-- also independently arrived at the idea. Did these men completely separately of each other invent the concept of natural selection? Of course not. They both built upon existing and well-fleshed out ideas in the developing field of biology and, after careful observation, arrived at similar interpretations.

So the notion of "independent innovation" is also ahistorical if the fertile bed of ideas and earlier technological development is not considered.

Basically, global and regional human history is a complex web-- a tangle, really-- of interactions and processes.

So what about the bow and arrow?

Bows and arrows developed in multiple places around the world. Preserved fragments of wooden bows have been found that date to between 17,000 and 18,000 years old in Germany. In South Africa, small stone projectile points of the appropriate size to work as arrow points have been found dating to well before that.

Because early bows were typically made from wood only, direct preservation is uncommon. Bow and arrow technology has been inferred from the size of projectile points found in some regions, which are small enough that they could have been affixed to the end of an arrow as opposed to a spear (too large a point and your arrow doesn't behave).

In fact, projectile point size is one of the primary ways the timing of the invention (or introduction) of bow and arrow tech has been inferred in many regions. A variety of studies-- including comparison of stone tool tech from earlier and later sites-- shows that arrow points generally don't get very large, and there's a pretty solid bimodal distribution pattern for the sizes of arrow points and spear / dart points (see below).

So are bows and arrows an example of independent innovation? Or diffusion?

They're probably the complex product of both.

Innovation

Bows and arrows weren't the first projectile tech. Thrown rocks and sticks probably long preceded any tools specifically made to be thrown. Even the earliest known wooden spears-- the Schoningen spears, which date to between 300,000 and 400,000 years old-- have been replicated and tested, and found to have been well designed to be thrown. So even with some of the earliest projectile tech, we still don't know the beginning of the story.

But at some point, someone or a number of someones came up with the idea that you could increase the power and distance of a thrown spear if you lengthened the lever used. The development of the spearthrower (or atlatl if you use the Nahuatl word) seems to also have occurred in multiple places around the world. To be effective, a spearthrower required a lighter spear-- a dart. The windup and sequence by which a dart is thrown has been shot on high-speed film, and it shows that a lighter, flexible dart absorbs energy before it's released, flexing like a spring. It then oscillates as it travels toward its target.

Obviously ancient people didn't have access to high speed cameras, but we have to give credit to ancient peoples, who mostly made their own tools and so were essentially experimental engineers.

So the idea of projectiles wasn't invented with bows and arrows. And an atlatl dart is more accurate with something on the back to add drag (which keeps the back in the back, allowing the projectile to travel farther and more accurately without tumbling in flight).

Atlatl / spearthrowers are known from practically every region of the world. Whether they were invented independently or diffused is still not really known. Probably a combination, depending on where you are.

So there was existing projectile technology in every region where the bow and arrow were invented.

Spearthrowers certainly came over to the Americas from Asia, so there's some diffusion there. But it goes along with population movement, rather than the idea or tech being introduced into an existing population.

So did the bow and arrow develop out of spearthrowers?

Possibly. It's not really entirely clear. Especially because there's a related piece of technology that was also widespread.

The bow drill probably developed quite early in human history. If you've ever used one, you know that if you don't keep a hand on the back of the drill (as shown in pictures) the drill will go flying off. Even more interesting, drills were usually tipped with small stone chips to allow them to cut / bore.

Could it be that in various places, during a drilling session (or firemaking session), someone's attention wandered, the drill flew off, and someone had the prehistoric version of a lightbulb go off in their head?

They already have spearthrower tech. Lightweight darts...

The bows for bow drills don't really flex like the bow from a bow and arrow, but suppose someone started experimenting. It might not be long before they hit upon the idea of basically a bow and arrow projectile system.

I'm not saying that this did happen, but the precursor tech is / was there in many areas for this to be a reasonable way to come up with the concept.

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The basic answer is that bow and arrow technology developed out of older tech, and probably was independently "invented" multiple times during human history. It also was likely adopted and refined by neighbors of the various people who came up with it in those areas where it was developed.

The reason that bows and arrows in the Americas, Europe, Africa, and Asia look similar is that there are only so many ways to create a handheld projectile delivery system, and the bow and arrow system follows fairly logically from earlier technologies.

The benefits of a small, lightweight system that can propel a lethal projectile at high speed are myriad, and in regions where this technology did develop, its adoption and refinement would have been rapid, basically a prehistoric arms race. So it can be difficult on a smaller regional scale to work out specifically where the tech developed, because it likely was adopted and spread so quickly among neighboring groups that our current means of telling time archaeologically-- radiocarbon dating, for example-- aren't precise or accurate enough to pin things down to less than a couple decades with any kind of reliability, and then only in the best of conditions.