I think the main UX obstacle is that it is unfamiliar – no-one does signups like that currently. But the flow does not need to be quite as bad, if you use "mailto:" links. In the happy case:

- user click on the link

- their email client opens, with the To:, Subject:, Body: fields pre-filled

- user clicks "Send"

- a few seconds later a sign-in link arrives in their inbox

`mailto` opens the Mail application on my mac, which I never ever used. I'd be surprised if that wasn't the case for most people.