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.
So long as this is occurring, the way the scheme works, unless you download every message, it could be determined which id belongs to whom. Once you determine that it is easy to track originating packets to that destination since it is not an onion system.
Additionally, without side channel communication ( in person or posting that address somewhere publicly ) it is impossible to anonymously communicate to an intended recipient.