That is not how the POP or IMAP protocols work. To make the system work with existing clients, the server needs to have the plain text when it is sending to the client. Surely this is also how they complied with valid warrants (wait until user logs in, then execute warrant).
Asymmetric encryption on the server means that an intruder can't read the content (except possibly for those accounts where the private key passphrase is still in memory) and nobody can search old mails until a user logs back in.
Um huh? What the heck are you talking about? Imap and pop can't tell if the body text of a message is plaintext or not beyond poop left by encoding /encrypting engines. Certainly my proposed system would allow plaintext messages disguised as encrypted messages to be sent to the end user, but so what?
I am not talking about how lavabit worked, I am asking why not design a system where the service provider need hold no private keys and thus have no way to comply with requests for keys. Yes, they can help the government track users and they can try to install malware on your machine, but fundamentally they don't know whether you're reading your email on a mac, pc, raspberry pie, or microwave oven.
Here's a pgp encrypted message. What can't I send by conventional email? (Or simply post on usenet, as my earlier post.) "They" can try to track every person who inadvertently downloads messages left for me.
Asymmetric encryption on the server means that an intruder can't read the content (except possibly for those accounts where the private key passphrase is still in memory) and nobody can search old mails until a user logs back in.