Sorry, forgot to add examples...
http://my-awesome-game.coolgamer.club/ is a demo I created,
http://ben-the-exorcist.coolgamer.club/ is a random one a user created... To get the email from Twitter, this is the authorization permissions I had to ask. The need for your email is so you can receive email signups directly there and I wanted to avoid the need to verify your address, so Twitter provided the solution... I think it goes without saying that it won't be posting on your Twitter profile and it won't spam. If you do not trust it, you could revoke access at any moment. But sorry no, there is no other way to sign up, at least for now...