Fix WordPress File Permissions Issue

My WordPress site has lost the ability to:

- Add a new plugin
- Update a plugin
- Add a new theme
- Edit robots.txt in the Yoast plugin

It seems like file writing has somehow become disabled. The common solution is to add define( 'DISALLOW_FILE_EDIT', true ); to the wp-config.php file, but this won't work in our Nginx environment since we don't run a wp-config file.

Requirements:
- Diagnose and fix the file writing issue
- Ensure all functionalities are restored

Ideal Skills & Experience:
- Expertise in WordPress and Nginx
- Experience with file permissions and server configurations
- Familiarity with security plugins and their impact on file operations

You will need to have access to the server's filesystem to check and change file permissions.