So the site was dead for a little over two weeks. I added support for comments, and after deploying the update to Heroku, the site crashed hard, showing “page not found” errors.
Usually, such failures are the result of my own mistakes, but this time all signs point to a change on Heroku’s end. As proof, I submit the fact that I tried deploying a fresh instance of the middleman-heroku-static-app buildpack and found that it failed in exactly the same way. Then, two days ago, the buildpack was updated in order to close out an issue titled “Get deploy running on Heroku”.
In the end, I used a different configuration to solve the problem, but the end result is the same. Hopefully the site stays stable for the forseeable future.