No Emails in Your Sent Folder? Check Your Filters!
Step by Step (Chrome): Gear Icon > See all settings > Filters and Blocked Addresses > Ctrl + F > type "Delete" or your email address > if a result shows "Action: Delete it," delete the filter.
This has been driving me insane, and I finally figured out the problem. Here is how I diagnosed and fixed it:
I checked the "Details" link at the bottom right of the Gmail inbox. I saw an "Authorized Application" accessing my account at the *exact same second* I sent a test email. Something was triggering a deletion.
Even after removing third-party app access, the problem persisted. It turns out a rogue filter (either created by an automated cleanup service or Gmail) was set to automatically delete anything I sent.
Perhaps this is an obvious solution to an easy problem, but at one point during my troubleshooting I wanted to throw my laptop out the window. IMAP/POP disabled, no third-party access, no nothing. Hope this helps.