Use 'Reference method' approach to allow 'Entity reference' view for selection

I agree, this would be nice. For anyone looking to achieve it via custom code though, pop something like this in a custom module:

use Drupal\Core\Form\FormStateInterface;
use Drupal\embed\EmbedButtonInterface;

/**
 * Implements hook_form_FORM_ID_alter().
 */
function MYMODULE_form_entity_embed_dialog_alter(&$form, FormStateInterface $form_state, $form_id) {
  $embed_button = $form_state->get('embed_button');
  if ($embed_button && $embed_button instanceof EmbedButtonInterface && $embed_button->id() === 'MY_BUTTON') {
    if ($form_state->get('step') === 'select') {
      $form['entity_id']['#selection_handler'] = 'views';
      $form['entity_id']['#selection_settings'] = [
        'view' => [
          'view_name' => 'MY_VIEW_NAME',
          'display_name' => 'entity_reference_1',
          'arguments' => [],
        ],
      ];
    }
  }
}

(Replace the various bits in capital letters.)