Especially if you work with sessions. I work full stateless for authentication and maybe I'm missing some obvious issues.
They started out with a login form that said that a link was mailed to you, but that you could also login with a password by clicking a button.
Now that has been changed to a login form where a choice between link-in-the-mail and password-based is offered in the form of two buttons.
I would guess they did this because it is really annoying to keep getting emails send to you, despite wanting to login with a password (it annoyed me!).
Ultimately, you will have to figure out if your audience is willing to put up with the extra friction you are introducing. The password managers in the browser makes logging in with a username and password trivial; having to go to a mail application, click the appropriate email, click the link, and delete the email, is more work than clicking in the username field and selecting the username from the dropdown box provided by the browser, which fills in the password as well.
Also, you are not adding any security per se, because you are delegating it to the user's email provider.
And when I think about the way you login to slack...
"Also, you are not adding any security per se, because you are delegating it to the user's email provider."
I think I do. In my case, I need an email and a way to recover the password. So, if the users email account is compromised, the account on my website is compromised too. Anything I do will only add potential breaches to the ones already existing in the users email provider ; not fix them.