Okay, so after a bit of research it turns out that their outgoing mail provider has been known to be misconfigured since at least 2015. They seriously just don't care if their customers can send email.

Don't get me wrong. I know exactly how easy it is to screw up a mail server config. Probably most people who have set such things up have messed it up, but keeping a broken config for years even when people are telling you exactly what's wrong seems pretty bad.