How to change the url of the archive page to be inside the custom post type

The has_archive parameter for registering the post type should work (untested):

register_post_type( 'news', array(
    ...
    'has_archive' => 'news/all-news',
    'rewrite' => array(
        'slug'       => 'news',
        'with_front' => false,
    ),
) );

Remember to flush the permalinks by going to Settings > Permalinks and clicking Save button.

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