To my great surprise, thanks to Tobias Sjösten, I found out that commenting was broken on blog.bigsmoke.us. I couldn’t pinpoint the exact problem, but it must have been introduced with some WordPress upgrade somewhere along the line. I never noticed it because it did work for logged in users. (If I must really guess, I suspect a silent ReCaptcha version compatibility problem.)
Upgrading WordPress and wp-recaptcha to their latest versions (3.3.1 and 3.1.4 respectively) seems to have solved the problem.