I.e. what Facebook does if you don't log in for long enough. Two days ago I got a pair of messages to the same address with links to completely bypass login and verbiage about how they'd seen I was having trouble logging in followed an sms message with the same to a phone number they're not supposed to be using. It looks a lot like phishing, but it comes out of Facebook's servers and they've done it to me before.

... and some sites use it instead of passwords, i.e. there are no passwords at all, only email links.

So SSO but you have to trust the email provider instead of another random SaaS

This sounds more or less like OTP.