So, I have reset the WSUS by following these instructions here
https://social.technet.microsoft.com/Forums/office/en-US/0119f29b-729e-469d-85b4-7d1e547ebe2d/how-to-completely-remove-wsus-on-windows-2012-r2?forum=winserverwsus
The clients are transmitting status reports again. It is still doing its first sync, but so far it's looking good again. Not sure what exactly caused this - maybe a corrupt database, maybe something else.
https://social.technet.microsoft.com/Forums/office/en-US/0119f29b-729e-469d-85b4-7d1e547ebe2d/how-to-completely-remove-wsus-on-windows-2012-r2?forum=winserverwsus
The clients are transmitting status reports again. It is still doing its first sync, but so far it's looking good again. Not sure what exactly caused this - maybe a corrupt database, maybe something else.