Pagination in category.php not functioning

if your aim is to restrict the category archive to your post_type ‘video’ and to 6 posts per page, do not edit category.php, rather use ‘pre_get_posts’ https://developer.wordpress.org/reference/hooks/pre_get_posts/

example code for your case:

add_action( 'pre_get_posts', 'category_post_type_video_custom' );
function category_post_type_video_custom( $query ) {
    if ( ! is_admin() && $query->is_main_query() ) {
        // Not a query for an admin page.
        // It's the main query for a front end page of your site.
        if ( is_category() ) {
            // It's the main query for a category archive.
            // Let's change the query for category archives.
            $query->set( 'post_type', array( 'video' ) );
            $query->set( 'posts_per_page', 6 );
        }
    }
}

virtually directly taken from https://developer.wordpress.org/reference/hooks/pre_get_posts/#targeting-the-right-query

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