The worst part is that I recall dealing with this exact same issue several years ago, but somehow failed to keep any notes about the little steps involved and other tidbits of relevant info.
So here are some quick notes just in case I find myself scrambling to fix the issue again in the future. Or if anyone else is dealing with the same task, here are some notes on Plesk webmail nonsense.
To disable storage of mail on the server while still allowing emails to route thru, click on the email address from the list and uncheck the “Mailbox” option.
Don’t forget to repeat the process for all domains on the server.
Setting a really high size limit is not a solution — it merely prolongs the issue.
In my case, the solution was to change the size limit via the subscription settings. I also disabled webmail just in case. Since then emails are flowing just fine.