How can I create a D7 View that groups and aggregates a "Rendered Node" field by date, grouping multiple instances for each grouped date?
You don't need the aggregation settings to get the result you want. Instead, I think you want the "Grouping field" option under the settings of "format".



Here is the export of a full View that I think yields the results you want:


$view = new view();
$view->name = 'date_group_test';
$view->description = '';
$view->tag = 'default';
$view->base_table = 'node';
$view->human_name = 'date_group_test';
$view->core = 7;
$view->api_version = '3.0';
$view->disabled = FALSE; /* Edit this to true to make a default view disabled initially */

/* Display: Master */
$handler = $view->new_display('default', 'Master', 'default');
$handler->display->display_options['use_more_always'] = FALSE;
$handler->display->display_options['access']['type'] = 'perm';
$handler->display->display_options['cache']['type'] = 'none';
$handler->display->display_options['query']['type'] = 'views_query';
$handler->display->display_options['exposed_form']['type'] = 'basic';
$handler->display->display_options['pager']['type'] = 'full';
$handler->display->display_options['style_plugin'] = 'default';
$handler->display->display_options['style_options']['grouping'] = array(
  0 => array(
    'field' => 'field_date',
    'rendered' => 1,
    'rendered_strip' => 0,
  ),
);
$handler->display->display_options['row_plugin'] = 'fields';
/* Field: Content: Title */
$handler->display->display_options['fields']['title']['id'] = 'title';
$handler->display->display_options['fields']['title']['table'] = 'node';
$handler->display->display_options['fields']['title']['field'] = 'title';
$handler->display->display_options['fields']['title']['label'] = '';
$handler->display->display_options['fields']['title']['alter']['word_boundary'] = FALSE;
$handler->display->display_options['fields']['title']['alter']['ellipsis'] = FALSE;
/* Field: Content: date */
$handler->display->display_options['fields']['field_date']['id'] = 'field_date';
$handler->display->display_options['fields']['field_date']['table'] = 'field_data_field_date';
$handler->display->display_options['fields']['field_date']['field'] = 'field_date';
$handler->display->display_options['fields']['field_date']['settings'] = array(
  'format_type' => 'long',
  'fromto' => 'both',
  'multiple_number' => '',
  'multiple_from' => '',
  'multiple_to' => '',
  'show_remaining_days' => 0,
);
$handler->display->display_options['fields']['field_date']['group_rows'] = FALSE;
$handler->display->display_options['fields']['field_date']['delta_offset'] = '0';
/* Sort criterion: Content: date (field_date) */
$handler->display->display_options['sorts']['field_date_value']['id'] = 'field_date_value';
$handler->display->display_options['sorts']['field_date_value']['table'] = 'field_data_field_date';
$handler->display->display_options['sorts']['field_date_value']['field'] = 'field_date_value';
/* Filter criterion: Content: Published */
$handler->display->display_options['filters']['status']['id'] = 'status';
$handler->display->display_options['filters']['status']['table'] = 'node';
$handler->display->display_options['filters']['status']['field'] = 'status';
$handler->display->display_options['filters']['status']['value'] = 1;
$handler->display->display_options['filters']['status']['group'] = 1;
$handler->display->display_options['filters']['status']['expose']['operator'] = FALSE;


And a screenshot of the rendered View: