Unfortunately in such a system, you have to download all emails, or request what you want in order to actual get the email you want.
The system either does not scale, or will leak information in order to remove the requirement of getting everyone elses emails.
Also, the government can collect -all- of the encrypted messages easily, and potentially eventually correlate who an anonymous id is based on their knowledge of who received messages on which day.
Basically, it sucks.