How to correctly edit permalink structures for both default and custom posts/taxonomies?

With @Caleb‘s help we’ve sorted this. The main problem was that the rewrite parameter on the custom taxonomies was not set to false. Doing this removed /articles/ from the permalinks to these archives.

We changed this:

$rewrite = isset( $taxonomy['rewrite'] ) ? $taxonomy['rewrite'] : array( 'slug' => $taxonomy['slug'] );
$hierarchical = isset( $taxonomy['hierarchical'] ) ? $taxonomy['hierarchical'] : true;

register_taxonomy( $taxonomy['slug'], $taxonomy['post_type'], array(
    'hierarchical'          => $hierarchical,
    'labels'                => $labels,
    'show_ui'               => true,
    'query_var'             => true,
    'rewrite'               => $rewrite,
));

To this:

$rewrite = isset( $taxonomy['rewrite'] ) ? $taxonomy['rewrite'] : array( 'slug' => $taxonomy['slug'] );
$rewrite['with_front'] = false; // new line
$hierarchical = isset( $taxonomy['hierarchical'] ) ? $taxonomy['hierarchical'] : true;

register_taxonomy( $taxonomy['slug'], $taxonomy['post_type'], array(
    'hierarchical'          => $hierarchical,
    'labels'                => $labels,
    'show_ui'               => true,
    'query_var'             => true,
    'rewrite'               => $rewrite,
));

This means I was able to adjust the permalink settings to achieve the results I required using the Category/Tag base fields and the Custom Structure field under Settings -> Permalinks in the WP Admin as usual.

The remaining issues were actually due to a setting buried in the Yoast SEO plugin. They have moved the setting which strips the category base to Settings -> Categories – > Additional Settings.

Huge thanks to @Caleb for the help here.

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