"What you're missing, I believe, is that the authentication is required at a certain time"
What you are missing, I believe, is that authentication is required at more than one moment in time, for more of one action or good.
You need to authenticate the voter to be who he says to be in order to avoid him voting twice. This you can get with this kind of system. But then you also need to authenticate that whatever that person voted is in fact what he voted. With paper you get this by the voter having the ballot in his full control from the moment the vote is decided (the moment he puts the vote within an envelope) till the moment he puts it in the ballot box and having an in-plain-sight chain of custody from that moment up to the moment the vote is counted. With an e-vote, how do you make sure whatever the voter decided is what got into the storing database without, at the same time, losing the voter's anonymity?
"...and the anonymity is required at a different, later time"
No. Anonymity is required along the full process. There *is* a moment when anonymity is also under risk with physical ballots and it is the period that goes from the voter putting the ballot in an envelope till the moment he puts it in the ballot box, and that's why this is done on a public place.
So, let's recap: on physical vote you gain both anonymity and authentication the Sherlock Holmes' way "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth." You don't really probe them, you just leave neither place nor time where/when the ballot can be tampered with or its content tied back to its caster. On a purely electronic system, there's no way you can offer neither such impossibility nor a positive proof that it cannot be done, as there's a single party -which is not to be trusted, owning the full process, from cast to storage.