drupal_render() leaves out the <input> tag when rendering a select element in a form table

Question

I have created a form with an embedded table for inputs, where one of the table columns contains a select field; the rest are textfields. The textfields all work fine, but the rendering of cells with the select field fails to create any <input> tag in its rendering. So naturally the form fails to act on any changes the user makes to this field.

In the following snippet, I have omitted a lot of data pre-processing. To give some context, I have created a $result array that has 4 or 5 rows of node data. Some of that get extracted to other variables to form the defaults for each form item.

This is the form.

function mymodule_walker_registration_form($form, &$form_state) {
  $form['table'] = array(
    '#theme' => 'mymodule_form_table',
    '#header' => array(
      array('data' => t('Walker\'s name'), 'class' => array('col-c3')), 
      array('data' => t('Date of birth'), 'class' => array('col-c4')), 
      array('data' => t('Age range'), 'class' => array('col-c5')),
    ),
    'rows' => array(
      '#tree' => TRUE,
    ),
  );

  foreach ($result as $walker_nid) {
    // a bit of data processing omitted: exctract various variables for use as defaults
    $form['table']['rows'][$row_number] = array(
      // Text field for the walker name
      'c3' => array(
        '#type' => 'textfield',
        '#default_value' => $walker_name,
        '#size' => 32,
        '#maxlength' => 32,
        '#required' => TRUE,
      ),
      // Text field for the walker date of birth
      'c4' => array(
        '#type' => 'date_text',
        '#default_value' => $walker_dob,
        '#date_format' => 'j/n/Y',
        '#required' => FALSE,
      ),
      // Select field for the walker age type ('scout', 'explorer', 'adult')
      'c5' => array(
        '#type' => 'select',
        '#title' => 'Walker age range',
        '#options' => drupal_map_assoc($age_ranges), 
        '#default_value' => $walker_age,
        '#required' => TRUE,
      ),
    );
  }

  // The submit buttons  
  // The 'Register team' button
  $form['e80progress_register'] = array(
    '#type' => 'submit',
    '#title' => t('Register team'),
    '#value' => t('Register walkers (get walker numbers)'),
    '#submit' => array('e80progress_team_registration_register'),
  );

  // The 'cancel' button
  $form['e80progress_cancel'] = array(
    '#type' => 'submit',
    '#value' => t('Cancel'),
    '#attributes' => array('onclick' => 'if(!confirm("You will lose any changes.  Are you sure you want to cancel?")){return false;}'),
    '#submit' => array('e80progress_team_registration_cancel'),
    '#limit_validation_errors' => array(),
  );

  return $form;
}

The hook_theme() implementation is nothing special.

function mymodule_theme() {
  return array(
    'mymodule_form_table' => array(
      // The renderable element is the form.
      'render element' => 'form',
    ),
  );
}

The issue comes in processing the table. I know that the select element gets this far.

function theme_mymodule_form_table(&$variables) {
  // Get the useful values.
  $form = $variables['form'];
  $rows = $form['rows'];
  $header = $form['#header'];

  // Setup the structure to be rendered and returned.
  $content = array(
    'header' => $header,
    'rows' => array(),
  );

  // Traverse each row.  @see element_chidren().
  foreach (element_children($rows) as $row_index) {
    $row = array();
    // Traverse each column in the row.  @see element_children().
    foreach (element_children($rows[$row_index]) as $col_index) {
      // Render the column form element.
      $row[] = array('data' => drupal_render($rows[$row_index][$col_index]), 'class' => array("col-$col_index"));
    }
    // Add the row to the table.
    $content['rows'][] = $row;
  } 

  // Theme the table and return.
  return theme('table', $content);
}

The only element that does not render correctly is the select cell. Visually, it is perfect. Functionally, it omits the <input> tag. This is how that cell renders:

<div class="form-item form-type-select form-item-rows-r1-c5"> 
  <label for="edit-rows-r1-c5">Walker age range 
    <span class="form-required" title="This field is required."></span>
  </label> 
  <select id="edit-rows-r1-c5" name="rows[r1][c5]" class="form-select required">
    <option value="Scout 12 years or older" selected="selected">Scout 12 years or older</option>
    <option value="Explorer">Explorer</option>
    <option value="Network">Network</option>
    <option value="Adult with DBS">Adult with DBS</option>
  </select>
</div>

How do I make drupal_render() or some other function to include the <input> tag into that cell of the table?