Should the plain text filter convert line breaks into p and div elements?
The answer to your question is no, it should not.

But I always want it to and so do my users. For textareas that do not require "fancy" formatting, we have still always wanted line breaks to carry over into the display.

I use this in my theme's template.php to add them to plain_text textarea fields:

/**
 * Implements template_preprocess_field().
 */
function THEMENAME_preprocess_field(&amp;$vars, $hook) {
  // Add line breaks to plain text textareas.
  if (
    // Make sure this is a text_long field type.
    $vars['element']['#field_type'] == 'text_long'
    // Check that the field's format is set to null, which equates to plain_text.
    &amp;&amp; $vars['element']['#items'][0]['format'] == null
  ) {
    $vars['items'][0]['#markup'] = nl2br($vars['items'][0]['#markup']);
  }
}