Drupal - How can I get both "body" and "summary" fields to appear in a new article?

I think you'll have to do it in a form alter hook as you suggest, I don't know of a way to do it in the UI.

Looking at the code for text summary fields the "Edit Summary" link and functionality are done completely through javascript, added in the form by setting an #attached property on the element, which includes a JS file (modules/field/modules/text/text.js).

All that JS file does is add that "Edit Summary" link so it would be completely safe to just get rid of it. This example uses the standard page content type:

function mymodule_form_page_node_form_alter(&$form, &$form_state, $form_id) {
  unset($form['body'][LANGUAGE_NONE][0]['summary']['#attached']);
}