There is a plugin that writes its own log files to the var/log/ folder, so I know the folder is writable by php, but no plugin_production-...log files are being written.
Log files are still being written on my staging copy of this site with the same code but running in dev mode.
Is this perhaps something to do with the mode I'm running in?
Please have a look at the default configuration: https://github.com/shopware/shopware/blob/5.4/engine/Shopware/Configs/Default.php#L263
The logging lever is changed according to your environment. You can change it via config.php and override the default-parameters.
btw: the debug plugin is outdated, you should use the profiler for debugging purposes: https://github.com/FriendsOfShopware/FroshProfiler