Including template pages within another template?

The function get_page() has been deprecated, don’t use it anymore, use get_post() instead. get_page_by_title() returns an object by default, you can change this by altering the $output parameter, which you can use like you did. Although get_post() can take an object as $id parameter I personally prefer inputing the ID, but that actually shouldn’t matter.

$page = get_page_by_title( 'About' );
$page_data = get_post( $page->ID );

echo $page_data->post_content;

What you’re describing – at least based on the code in your question – should have nothing to do with templates or the get_template_part() function, because the post_content you’re getting is independent from that. The case would of course be different from that, if what your are missing from those pages you talk about isn’t actually part of the post_content, but your question isn’t indicating that.