Can confirm that, on my Mac at least, Chrome and Firefox behave as @DL6ER says. This is why I was using dig to test, which seems to do fresh DNS queries.
Aha! I had been setting the cron rule using sudo crontab -e when logged in as pi on the Raspberry Pi running pihole; my assumption was that the sudo would mean that I was editing the crontab for root, but I now think I was wrong about that.
Editing /etc/crontab directly and specifying root as the user for the cron rules seems to have worked. Hoorah! Thanks, @DL6ER