I thought I had disabled and deleted Jetpack from all client sites using ClassicPress, but apparently I missed one. Or perhaps the client had added Jetpack on their own.
At any rate, adding a new post (including custom post types) or page was impossible, because clicking on the add button presented the user with a completely blank page. Disabling Jetpack immediately resolved the issue. I didn’t look any further into the exact conflict, so I have no other details to share.
Sharing now in case someone else runs into the same issue.