I am a part of Jessica's livestream team. I'm sorry for your frustration! The music laws issue has to do with how long we are allowed to leave up the Dance2Fit videos. This is a very real issue that many people choose to ignore at risk of getting sued, which is why you don't see other people adhering to it. The archiving function is not up to us at all, unfortunately. Since we are on such a large platform that has to communicate with more than just a website (several apps as well), the transcoding process takes longer than we would like. Vimeo assures us that they are working on making it faster.
You should be able to start a live class from the beginning if you tune in late just by pulling the time bar back to the beginning. I will check this function on the next class to make sure nothing has changed.