If Paperless-NGX is one of the things you self-host; it has options to import emails based on your specified criteria, then you could have it delete each piece of mail it imports. You can also just have it move mail to folders on the mail server, or just tag/flag mail instead of deleting it. (for you to then manually delete at your leisure)
I use this to automatically import receipts, bills, work documents, and any other regular mail instead of dealing with it manually every week/month.