Jumping in the middle here, but just to clarify a few things:
The "Web mail MTA Hostname:" in the Admin UI is the MTA server that end-users running the web client will use for outbound emails. If you have a single server, this is properly set to localhost, but you can also enter the FQDN of the single Zimbra server.
In a system where there is a separate MTA server apart from the mailbox server, users logged in via the web client will have their email sent from the mailbox server to the MTA server, which will then route the mail to its next hop (either a Zimbra mailbox server in the farm if the domain is in the farm, or the MX listed for the recipient's domain elsewhere). Since this setting is mailbox-server-specific, you can do things like have multiple MTA servers and configure different mailbox servers to use different MTA servers. In this way, one MTA server failing doesn't block outbound email for all users, and the mailbox servers pointing to the failed MTA can be redirected to the working MTA.
The setting in the Admin Console "Relay MTA for external delivery:" is useful only on MTA servers and if non-blank is the sole server to which the Zimbra MTA server will hand off its email.
If you have a small Zimbra installation and your ISP provides SMTP services for customers on its network, it can be helpful to put in your ISP's SMTP server here. Although you will lose some ability to track non-delivered emails, if you have a compromised user your Zimbra server won't get blacklisted.
Re the daily mail report not being delivered, two things I haven't yet seen mentioned in this thread.
First, have you checked your Junk Folder? It is not uncommon that Zimbra flags the Daily Mail report as Junk...
Second, please post your /etc/hosts file here and/or double check it against the documentation to make sure the localhost entry is correct.
Hope that helps,
Mark
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