As per confirmation from @Jacob Peattie, my understanding is now this (if it is helpful to future readers).
AJAX calls to a WordPress plugin method exist in a totally separate context (separate requests) from the object instance created when a WordPress plugin’s PHP files are interpreted when a page is loaded.
Therefore, a property that is given a specific value only during runtime (for example, $this->foo), while being visible during a given session, is not guaranteed to be visible to an AJAX call. The AJAX call accesses the object’s property as it is set on new object instantiation OR if it were set or changed via and during the AJAX call.
To make a dynamic data change visible between JS and PHP files, even if belonging to the same plugin package, likely requires another storage method.