Funnily enough, it seems to be working now with the “Day and name” option! And now I know why it didn’t work earlier: if you haven’t chosen a posts page in the “Settings >> Reading” section, it displays all the pages on one page instead of the 404 page. When you choose a posts page from the dropdown menu in “Settings >> Reading”, the error pages start to work properly.
However, it’s a bit annoying to have a page for the posts if you currently don’t need a blog, but at least the error page works now.