Given that we don't have written records from these periods, because writing had not been invented at this point, it's hard to imagine what kind of evidence one would be even looking for.

But that's not really what I want to address. What I want to address is: would such people even have a concept like "extinction"? It's hard to say, given the lack of specificity about what their beliefs and knowledge were (owing to our lack of written records!), but I would suggest that from what we know of later people, people who did write down their views, the answer is "probably not."

The concept of "extinction" is much more modern than most people realize. To even imagine it, you have to imagine that the world and its resources are finite and extinguishable, something which is very difficult to do if you do not have some way of systematically recording those resources, and if you have a belief-system that involves any kind of deities being responsible for providing natural resources (e.g., if you think God is ultimately in charge of what animals exist on the planet, then it can be hard to imagine that He would let them "run out").

The first animal that people seem to noticed having been totally depleted from the world was the dodo bird, and even that required very specific circumstances: a very unusual bird having a very limited habitat; people visiting that habitat and making records as to the existence of this unique bird; people eventually noticing that they found no further evidence of the bird in this habitat over the course of many decades. And even then, they had to believe that there weren't a few more of these birds hidden away in the habitat, or in some other habitat in the wide, wide world that they knew they had not fully explored. The modern understanding of extinction dates from the late-18th century and is credited to the scientific work of Georges Cuvier, and is related to the discovery and study of fossils. But many fossils of extinct species had been found prior to this, in the early modern period, and many of the brightest minds of that age had dismissed the possibility of extinction altogether on theological grounds (they believed they might be either not actual animals that had lived, or that there might be examples of those animals still found in places not yet explored). So even finding fossils doesn't guarantee you believe in extinction!

How could a pre-modern peoples really form this kind of idea? You could imagine them passing down, in oral traditions, stories about interactions with animals that no longer were found, or artwork of such animals. But how would they know those animals were truly and finally gone from the world? Much less conceive of such a disappearance as being the result of human activity? It is just hard to fathom, given the difficulties that even "modern" and "scientific" people had making sense of it.

And even if some individual or group of them had figured this out — how would they wage a campaign to protect a species? You are talking about a world before states. Sure, some group could avoid hunting some animals. But that isn't much of a "campaign."

To be sure, one might find interesting examples of pre-modern peoples having what we would today consider ecological sensibilities. We know that many pre-modern peoples had complex relationships with, and understandings of, the ecological environment. Many clearly understood that over-hunting could be a vice (and this kind of thing is reflected in many indigenous belief systems, which admonish those who would hunt more than they need, or label certain species as totems that cannot be hunted, etc.), as could over-foraging and over-harvesting and so on. But these are different from thinking about extinction, I would argue, which is a far more tricky concept to put together, and required a number certain technical, ideological, epistemological developments to emerge.