This is actually wrong, you can choose to have your streams recorded and placed in the 'Videos' tab on your profile where others can watch them forever.
Couldn't you simultaneously do the twitch stream, but also record and host the shows elsewhere, if you wanted that library of past shows?
I see this with a lot of folks - they stream on Twitch and then the audio shows up on Mixcloud/Soundcloud, and sometimes the video on Youtube later on. You get the fun of watching it live, but then also enjoy replays later on.