if (!empty($errors)) {
    $status = ['status' => false, 'errors' => $errors];
} else if (FALSE === $wpdb->insert('people', ['name' => $_POST['name']], '%s')) {
    $status = ['status' => false, 'errors' => ['Database error: ' . $wpdb->last_error]];
} else {
    $status = ['status' => true];
}

echo json_encode( $status );