Prior to about 2013, many class methods lacked even access modifiers which made the `@access` notations that much more useful. Now that we've gotten to a point where the codebase is more mature from a maintenance perspective and we can finally remove these notations. Notable exceptions to this change include standalone functions notated as private as well as some classes still considered to represent "private" APIs.
See #41452.
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`$context` is a full path to the directory that is tested for being writable. A path shouldn't be a boolean value.
This also updates `WP_Upgrader_Skin::request_filesystem_credentials()` and `Automatic_Upgrader_Skin::request_filesystem_credentials()` and adds missing docs.
Props DrewAPicture, ocean90.
Fixes#37412.
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