Deleted (but not cleared) voicemails not visible on iPhone 5
I solved my problem by resetting the network settings from Settings -> General -> Reset -> Reset Network Settings (this course of action is actually recommended in the igeeksblog post I linked, but for the circumstance where Clear All is disabled rather than for the particular circumstance my phone was in). After this, I was able to call my phone and successfully leave a voicemail. I also tested out deleting a new voicemail, and it went into a Deleted Messages queue that is now visible at the bottom of my Voicemail Tab, as usual.