We could ask /r/askanthropology, but I suspect every culture has heroes. And if it is not "every culture," we can at least agree that most cultures have heroes. So we can also imagine that the prehistoric antecedents of the Bronze Age also had stories of heroes.

The Bronze Age represented that time that was the first to be subjected to extensive writing; stories were written down, and not surprisingly many of these stories involved the sort of great heroes who were certainly the subject of entertaining stories long before writing. It isn't that "Bronze Age cultures produced classical hero figures," as much as it is that authors had emerged who could record heroic accounts set in a Bronze Age frame.

In short, we need to turn your paradigm on its head and understand this not as a period of the production of myths but rather as a period of recording of stories.