Okay, I’ve done some more digging. I found out that SOME members-only pages were working and some were not. The common denominator was “Page Builder by SiteOrigin”. When I finally got a lead, I was able to fix it quickly. Disregard.
I am also having this issue: non-members can see members-only content. All the right boxes are checked, so don’t bother asking if they are. I have also not installed any caching plugins. I was sure that it worked a while back, but no longer. Several plugins and the theme I’m using (Zerif Pro) have been updated in the meantime, so I’m wondering if maybe the issue resides somewhere in there.