I understand your request, and I understand your pain, but I'm not sure that it is up to Mailman to make up for deficiencies in cPanel and various Mailman hosting services. I really think it's up to cPanel to provide ways for hosted users to access the Mailman file system through cPanel. I understand that this raises many other issues in a hosted environment, but I had a conversation with a cPanel developer at the last PyCon, one result of which can be seen at <https://mail.python.org/pipermail/mailman-users/2013-March/074860.html> so there may be some movement in this direction with cPanel.
I also recognize that even if a perfect solution were available through cPanel, that wouldn't help customers of other, non-cPanel Mailman hosts.
In any case, even if we provided within the Mailman list admin GUI a way to download and upload, say a list's config.pck which is the only thing that contains all the list settings and member data (assuming the default MemberAdaptor is used), I envision a user grabbing a cPanel config.pck and uploading it to a non-cPanel Mailman or vice versa and disaster ensuing when all the web UI returns is a "we hit a bug" screen with no way to even restore the prior config.pck even if the user had been careful enough to make one.