Update table wp_term_taxonomy.count after INSERT INTO via SQL

You can just run the wp_update_term_count_now php function after doing this:

https://developer.wordpress.org/reference/functions/wp_update_term_count_now/

Which calls the private _update_post_term_count

https://developer.wordpress.org/reference/functions/_update_post_term_count/

If you look at the source of that you will see how term counts are updated (which is done when you save a post on backend):

function _update_post_term_count( $terms, $taxonomy ) {
    global $wpdb;

    $object_types = (array) $taxonomy->object_type;

    foreach ( $object_types as &$object_type )
        list( $object_type ) = explode( ':', $object_type );

    $object_types = array_unique( $object_types );

    if ( false !== ( $check_attachments = array_search( 'attachment', $object_types ) ) ) {
        unset( $object_types[ $check_attachments ] );
        $check_attachments = true;
    }

    if ( $object_types )
        $object_types = esc_sql( array_filter( $object_types, 'post_type_exists' ) );

    foreach ( (array) $terms as $term ) {
        $count = 0;

        // Attachments can be 'inherit' status, we need to base count off the parent's status if so.
        if ( $check_attachments )
            $count += (int) $wpdb->get_var( $wpdb->prepare( "SELECT COUNT(*) FROM $wpdb->term_relationships, $wpdb->posts p1 WHERE p1.ID = $wpdb->term_relationships.object_id AND ( post_status="publish" OR ( post_status="inherit" AND post_parent > 0 AND ( SELECT post_status FROM $wpdb->posts WHERE ID = p1.post_parent ) = 'publish' ) ) AND post_type="attachment" AND term_taxonomy_id = %d", $term ) );

        if ( $object_types )
            $count += (int) $wpdb->get_var( $wpdb->prepare( "SELECT COUNT(*) FROM $wpdb->term_relationships, $wpdb->posts WHERE $wpdb->posts.ID = $wpdb->term_relationships.object_id AND post_status="publish" AND post_type IN ('" . implode("', '", $object_types ) . "') AND term_taxonomy_id = %d", $term ) );

        /** This action is documented in wp-includes/taxonomy.php */
        do_action( 'edit_term_taxonomy', $term, $taxonomy->name );
        $wpdb->update( $wpdb->term_taxonomy, compact( 'count' ), array( 'term_taxonomy_id' => $term ) );

        /** This action is documented in wp-includes/taxonomy.php */
        do_action( 'edited_term_taxonomy', $term, $taxonomy->name );
    }
}