Some things that you said make a lot of sense. But I think the buttefly symbolism is more because of the butterfly effect: how initialy tiny actions can make for a big change on a system. The butterfly logic is probably what Kate uses to amplify the Pattern to begin with.

I also believe the players is just a person coming to the town some time after the events :) Now “lights” (dead characters, like lizzie, kate, etc) are trying to show the player their story.