You download the LifeGen file from the LifeGen Discord server. On the server, they have a channel named "how-to-download". That's where the tutorials are located.
The file link/download is located in "game-announcements".
Yeah, it does the same thing for me sometimes. I think it might just be a bit buggy? For me, after waiting a bit (like 5-6 minutes) and the screen is still black, I try to reload the page and wait again. Usually the page starts working. But sometimes it also doesn't. So you've just gotta play around with the page.