How can I get Civimail to deliver emails instead of bouncing them?
Issue solved!

Somehow I had changed the setting in the (one) mail account in Civimail (Administer -> Mail accounts).

The "Used For" field had been set to "Email to activity processing" instead of "Bounce Processing".

So there was no bounce email assigned so the the send was being rejected. Setting the "Used For" to "Bounce processing" fixed the issue.