Earlier quoted context omitted.
I hate the email login link stuff, and any site that has it as the only option, I move on unless absolutely forced to use it. Especially a pain if I'm trying to login on a device without my email.
You don't need to have email on the device you are logging in from Can take out phone and click link
That assumes you have that email account on your phone. Also that you have your phone with you, that your phone has signal / wifi, there's no delays, no greylisting, no spam filtering that might catch the email, ...
There's just too many simple ways it can break down for it to be a good system.