To embed a form in a renderable array, you use:

    $form_class = '\Drupal\my_module\Form\MyForm';
    $build['form'] = \Drupal::formBuilder()->getForm($form_class);
Which is similar to your code.

But the problem is of a different nature. You are extending a special controller, one that returns a renderable array of `'#type' => 'page'`. It uses the page template to render the output and that template does not contain the `{{ layout_control }}` variable to output your form. If you use `$page['header']['layout_control'] = ...` to add the form to the render array, it should show the form in the header region of the layout demo page.