So your only choice is to access your (personal) email on a device where you may not want/be allowed to.
I think password managers are great but the service I see them providing that has the most benefit is to reduce the number of passwords the user needs to actively remember. The single sign on approach has it's weaknesses but the idea itself is pretty sound, you should legitimately trust facebook or google or whoever to know more about secure authentication than some random site... the fact that this service comes with a privacy leak and implies greater online presence tracking is terrible, but the core idea is sound. It'd just be nice to have a neutral party doing it.
The problem is the lack of an alternative which is how this whole thread started - their approach allows you to choose between logging in via a third party, or logging into your email on the same device. It doesn't let you log into their website directly.
I wouldn't mind if they emailed you some kind of time-limited one time token you could enter on the other device, rather than using a conventional password. But all they supply is a lengthy link, which can only practically be opened on the device that has access to your email.