Just a few off the top of my head:
1. Easier for email hackers to detect sites where you have logins. (Controlling someone's email usually means controlling most/all logins, but it takes some digging to get a good list of vaulable logins. With this solution, most of the list is on the first page or two of the inbox.)
2. Harder to detect being hacked. (Previously, a hacker with email access would have to reset your passwords, and you will notice that at least some passwords have changed. Now the hacker just has to delete any incoming authentication emails after reading them.)
3. Losing a job becomes potentially more catastrophic. (Hope you didn't associate too many passwords with your work email, because IT wiped your account while security was escorting you out the door. And before you say only dumb people use work email for personal accounts, consider that part of the idea of this nopassword system is to help "dumb people" who fail to (for example) use password managers.)
4. Your email provider now has a nice easily mined record of what sites you log into most often. But, hey, I'm sure we can all trust Google not to use that information in a terribly creepy manner, right?