Then a new and huge list of security problems arises when you have to bother the user with getting a new code every time if they have the sense of closing their browser and cleaning their cookies each time they close their browser (which could be as often as whenever they leave their computer); the fact that loosing control of a single email makes you lose control to the account in every site using this system, which beats the idea since that email is most likely password protected anyway; etc, etc.

In a nutshell: "In most cases you won't need to do this often" is a HUGE fallacy. It depends on the security rules you work/live by. Plus, it would make it really annoying to use if on top you're using TOR.

Yes, passwords need to be fixed. They are weak, problematic and a security cheddar cheese. It is why we are now implementing two factor authentication. Changing the "fixed password" strategy to a "random and time limited password" strategy isn't exactly solving more issues than it raises. Again, from a security-wise stand point.

May be if this was implemented with something different than your email. Like, for example, a bank tokens or cell phone verifications... which, again, are part of a two factor authentication because by themselves they would be too easy to break.

Think about the following scenario: You use X site with this email auth system and, for example, Thunderbird. Stand up and go to the bathroom or a meeting or whatever without locking your computer. Presto! I won't even need to guess a password and get access. Of course getting access to X site would be the least of your worries in that example, but it illustrates the point I'm trying to make.