It hadn't even occurred to me that there were alternatives - many thanks for
the pointer!
Installed it, similar results (a 404 this time) but a quick search revealed
that there are still a few manual configuration steps, for some reason not
included in the package:
sudo ln -s /etc/adminer/apache.conf /etc/apache2/conf-available/adminer.conf
sudo a2enconf adminer.conf
sudo service apache2 reload
After which it worked like a charm.
I won't mark this as 'solved' because it would still be good to fix
phpMyAdmin - anyone have any more ideas about what might be wrong with it?