The default setting of the "Save past broadcast" option is setturned offby Twitch. Twitch allows users to save the broadcast but due to the high influx of the Twitch user, this setting is set as "turned off" to make sure uninterrupted services to users
No, Twitch streams did not save automatically, Twitch has millions of active users on daily basis. If this option is set as automatically then twitch needs huge storage serversto do this task and this might disturb the smooth streaming.