Using the page slug the correct content shows – but I actually think this might be the problem. The email that was showing had the same subject line as the page that was supposed to be displaying, so I think this created trouble for the plugin.
In effect, a message was sent via Contact Form 7 (and saved with Flamingo). That message had exactly the same subject line as the slug of the page being referenced by Insert Page – this, I’m guessing, resulted in the saved message being shown instead of the page.
I’m going to try switching Insert Pages to the Page ID mode instead of Page Slug and see if that solves the problem.