Had a mouse issue and tried logging into my Windows 8 Guest account to see if that could jog something, then it occurred to me that I could use the Opera installation in the Guest account to test this.
The Opera Guest profile is a clean profile: no extensions and only the default bookmarks.
Same issue with this site not staying logged in between Opera sessions.
Will contact the site's tech support to see if there may be some weird browser-specific login issues (since Chrome stays logged in between sessions).
Will probably start with a clean slate anyway since some of the notes in my Simple Notes extension became corrupted (changed from plain text to text with embedded HTML) after years of trouble-free usage. I'm manually weeding through and copying my most important notes to make sure I don't lose data doing a complete reinstall.