The solutions here are good. There are issues, which compromise your backup fidelity
- bcc loses original recipient lists
- ISPs are notoriously unreliable re: email delivery - you will have some number of lost messages
If these are not concerns, then no worries.
The solution for more demanding needs and better fidelity of course is to either use professional services, or setup and maintain your own mail server. For multi-system simultaneous mailbox view and access, IMAP is wonderful.