Wordpress Plugins are stored both in the filesystem and in the database and need to match. Same for the Wordpress Core when an update occurs as WP updates the PHP files locally for the upgrade process. The AWS guidance for a multi-webserver Wordpress setup has included EFS for this reason for years now. You can store uploaded media in S3, but that is a performance optimization that doesn't solve the previous issues.
Not OP but the obvious use case for Wordpress is the install directory itself. Deploy to one server and have all servers sharing the EFS get the update at the same time.
There's a cache. Even if you have to restart some process or purge some cache, this setup makes sense to me for applications that can update their own code (e.g. install/update plugins).