LMTP is for delivering to an IMAP (or JMAP) mailbox, generally. We also have a plugin for that. But it has certain flaws for exposing it to the internet (I just forget what they are right now).
For context, I'm the author of Haraka, a popular open source SMTP server.
SMTP servers don't necessarily store anything. They have two responsibilities: Receiving and delivering email. What they do with the emails they receive is up to the configuration.