The first does not follow from the second. Historically, incentives on human behavior have tended much more towards collaboration than competition. If all human behavior was framed as a competition, we would not have been able to construct communities.
Can you cite the pop-sci TV show you got that from? You've got it exactly the wrong way around.
Evidence of deadly human-on-human violence mostly starts showing up with the Neolithic period, before that humans didn't go out of their way to kill each other as much. It looks a lot like you need to be settled and amass possessions to give people a good enough reason to kill each other. That's recent behavior as far as our species is concerned.
When you say “fairly new” what timeframe are you thinking about?
Just because we have a ton of evidence of humans collaborating as far as historical artifacts go back. But then of course with the appropriate perspective almost everything is fairly new. Would you call Göbekli Tepe fairly new for example?
After all 9000 BCE was just yesterday compared to the invention of multicellular life for example. Which is also massively about cooperation.