Pagination doesn’t work with custom post type

I recently had a site with a custom post type and the pagination was not working because the post type slug was the same as the page slug (news and news).

I wasn’t using the archive page to display the post loop so I disabled the post type archive in the file I initiated the custom post type:

register_post_type( 'news',
    array(
        [...]
        'has_archive' => false,
        [...]
    )
);

And then I added this rewrite to functions.php:

// prioritize pagination over displaying custom post type content
add_action('init', function() {
  add_rewrite_rule(
    '(.?.+?)/page/?([0-9]{1,})/?$',
    'index.php?pagename=$matches[1]&paged=$matches[2]',
    'top'
  );
});