How to get category pages to look like a certain archive page?

You could achieve this with a filter on category_template. This would go in your theme’s functions.php file. See the comments in the code for an explanation of what it’s doing:

function video_category_archive( $template="" ){
    // your parent video category ID
    // ( change this to your actual video category ID )
    $video_cat_id = 42;
    // get the ID of the current category being viewed
    $cat_id = get_queried_object_id();
    // get the IDs of the ancestors of this category
    $ancestors = get_ancestors( $cat_id, 'category' );
    // if this category is a child of the video category,
    // or if this is the parent video category,
    // load the post type archive template
    // instead of the normal category template
    // ( change archive-video.php to the name of the template you want to load )
    if( in_array ( $video_cat_id, $ancestors )
        || $cat_id == $video_cat_id ){
        $template = locate_template( array( 'archive-video.php' ) );
    }
    return $template;
}
add_filter( 'category_template', 'video_category_archive' );