I don't mean to imply that just visiting the link should be enough to complete a login. That's a GET and there's a LOT of issues with doing anything important on GET. Just "do something on a different machine, then automatically complete login on the one logging in", and magic links to trigger that flow are a rather straightforward option.

There's no reason at all that it has to all occur on the same machine, and many reasons why attempting to require that doesn't work out in practice even when it does happen on the same machine.