The Broadway lotteries use social media authentication so they don't have to store and maintain user information (emails, usernames, passwords). It's easier and more secure for developers to use an integration with a social media site like Facebook/Linkedin/Twitter than to have to deal with properly sending/receiving/storing passwords. This comes with the added bonus of being able to push "Share your entry on Facebook/Twitter" incentives to users to serve as advertisement for the show, but the primary reason they use these social media sites is actually because they don't want to be responsible for your data (at least not your passwords).
SocialToaster is just so janky looking. I really wouldn't trust it to do proper username/password authentication even if the option was there. Using an empty Facebook/Twitter would essentially give you two factor authentication, which would keep your payment info more secure. That's how I've always done it. You can tell LuckySeat is a lot more well maintained, and they offer username/password login, no social media required. (But LuckySeat seems to be connected to the whole Jujamcyn/SeatGeek universe, which has it's own ticketing flaws.)
I did a few in person lotteries when I lived in New York City, and while they were fun experiences, I can't say I really "miss" the freezing temps or long waits before going home empty handed. The digital lotteries were a great way to see shows on the cheap. Plus, you can enter multiple in one day! Not sure how easy that would be in-person.
