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.

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.