On a hunch, I tried unapproving the comment and then approving it again. This seems to have worked. So I imagine the problem was that, the first time I approved it, the email hadn’t yet been sent, so that confused something.
I cleared my browser cache and nothing changed (or did you mean something else by my “website cache”?). Again, the file public_html/wp-content/endurance-page-cache/<name-of-post>/_index.html is in what appears to be an inconsistent state; I don’t see what that could have to do with caching (unless there’s some strange other form of caching that I’ve never heard of, of course, in which case please enlighten me).
That was indeed the issue. I had thought those single quotes looked suspicious, but figured they must be there for some inscrutable reason of WordPress’s own. I wonder how they did get there; I’m pretty certain I didn’t type them in. Thanks!