ImprovMX does this just fine
We did, tried something "original" in the login part, by offering a "one time login link" to our users instead of the standard flow. To be honest, we believe this was a bad idea.
The flow in itself is good and works fine (supposing email delivery works), but as @simonw perfectly said it, "resist the temptation to innovate around login!".
Offering that original "one time login link" is a frequent cause of support request because some users doesn't receive it, doesn't know what to do or how. They are not used to a change on login and this causes more troubles down the road than a standard flow.
That's why we later on decided to add the standard "password" login and tried to reverse that original flow. We still have users that have accounts with no password set though (still connecting via the original "one time login link").
Especially a pain if I'm trying to login on a device without my email.
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.
We're used to one way of working and have our setups for that. We don't want something different but not because it's worse. It's just different.
And even more, when you have a problem with your password, what do you have to do, yes, email for the most part.
Plus I don't know who's using that to help adtech build their cross device mappings