Improve the compatibility of WP_Admin_Bar with PHP 8.2

[antonvlasenko]

The WP_Admin_Bar class currently uses the __get magic method to return values for some of its properties.
These dynamic properties are not supported in PHP 8.2 and above.
Therefore, WP_Admin_Bar should be refactored to remove the dynamic properties while ensuring that backward compatibility is preserved.

[antonvlasenko]

I'm working on a patch for this issue.

This PR aims to improve compatibility of the WP_Admin_Bar class with PHP 8.2 and above by removing dynamic properties while making sure that this won't break backward compatibility.

Trac ticket: https://core.trac.wordpress.org/ticket/56535

[antonvlasenko]

Duplicate of #56876.

This issue is closed in favor of https://core.trac.wordpress.org/ticket/56876, and the related changes have been moved from ​https://github.com/WordPress/wordpress-develop/pull/3216 to ​https://github.com/WordPress/wordpress-develop/pull/3535.
Why?
This ensures that all the classes with the self::$compat_fields property get fixed in the scope of a single PR.

https://core.trac.wordpress.org/ticket/56535#comment:5