I reached out to Sony, who responded with the following note: "We are currently working on a fix that will enable archiving of Until Dawn Twitch streams as this feature was unintentionally disabled. We are humbled by the community reception of the game and are excited to see fans sharing experiences on Twitch and YouTube. We apologize for the inconvenience and will provide an update as soon as the issue has been resolved."
I mean, the move kind of made sense. It would be similar to streaming a movie that just released on Blu-ray -- the game is basically an interactive film, even more-so than a lot of other recent adventure efforts.