It's almost certainly the other way around since Battle Royale was published in 1999 and the Battle Royale film came out in 2000. Both probably take some inspiration from Lord of the Flies
But why did the genre blow up only now? I think that the mass culture projects massive amount of anxiety present in todays societies, and it is also visible in video games. I don't see how a bleak and dystopian game of dog-eat-dog, everyone on its own, 1 player wins and the 99 others lose, would be so popular in say 1995.
I think you're looking for a deep meaning that is not there. Video games are fun, period. The 12-yos that play Fortnite couldn't care less about dystopias.
Internet was hardly available in the year 2000. It takes a lot of bandwidth (and probably CPU too) to run a 100 players FPS. It wouldn't do well before broadband got ubiquitous, almost a decade later.
That doesn’t change the fact that BR predates the hunger games (though the author claims to not have heard about BR when she wrote the hunger games).
BR is supposedly about how the old of japan are metaphorically sacrificing the youth. The hunger games was (according to an interview) about reality tv meets the war in Iraq.
