How can I override the HTML of options on the user edit page so that they do not appear as collapsible fieldsets?
The easier way is to set the #collapsible property to FALSE.
As far as I know, when #collapsible is set to FALSE, #collapsed is not used anymore, which means the fieldsets are shown as visible; if that is not the case, then set also #collapsed to FALSE.

Clearly, this works only for fieldsets; for other form fields, those properties are not used.

As side note, on Drupal 8 collapsible fieldsets are implemented using "details" as form field (at least from the Drupal core code); if you are going to loop through the elements of a form to check for which ones the property should be set, the code should be similar to the following one, for Drupal 8.

foreach (element_children($form) as $id) {
  if (!empty($form[$id]) &amp;&amp; ($form[$id]['#type'] == 'details' || $form[$id]['#type'] == 'fieldset')) {
    $form[$id]['#collapsible'] = FALSE;
    $form[$id]['#collapsed'] = FALSE;
  }
}