They've managed to already for it for NFL and MLB rights, and there's plenty of cheaper rights out there and up for grabs. Netflix didn't start producing original content to begin with, but Twitter could ramp their way up to obtaining a lot of the crown jewels of sport.
Their live streaming was all about content negotiation and marketing which I think they did a poor job of. The actual streaming was handled by MLB Advanced Media which is far and away the leader in this space. Twitter should just acquire them if they want to get into this space.