Eventually I learned it was because Radix, my TLD registry, had noticed my domain on the Spamhaus Domain Blocklist. Who knows why it was added; I definitely hadn't been sending spam. Radix, being proactive, placed a 'serverHold' DNS status on my domain name, preventing it from resolving.
Luckily Spamhaus has an automated form for dealing with false positives. Eventually Radix removed the status. (Though, a week later it was back up, for no reason. Had to badger them for over a week to get the status removed again. Would not recommend them.) It's very lucky I wasn't trying to do anything with that domain like run a business, or rely on @<domainname> addresses to work.
TL;DR the problem is definitely not overstated.