I'm trying to share a private YouTube Live broadcast with a Gmail alias that has 200+ people on it.
Will YouTube allow sharing with a mail/distribution list, or do I have to enter each email one by one?
YouTube needs to authenticate the individuals watching the video and for that they use the person's Google account. So yes, if you make your stream private, you'll need to enter each address individually.
However, on that scale, you may want to use an unlisted stream instead and trust that your mailing list members won't share it. Or use a different service that has domain restrictions or password protection or so (eg. Vimeo Live).