Why isn’t my custom widget outputting any content?

Using the_widget() denies those $instance variables being set, which they normally are when you stick it in a Sidebar in the back end.

Try something like this:

the_widget('bannerWidget', array(
    'title' => 'WordPress rocks!',
    'text' => 'Hello World.',
    'textarea' => 'Some content for the textarea'
) );

the_widget is for hard-coded widgets. To use the one placed in your custom sidebar, replace the_widget() with this:

echo '<ul class="sidebar">';
dynamic_sidebar( 'test_sidebar_1' );
echo '</ul>';

dynamic_sidebar is for displaying all widgets inside of the specified sidebar (passing the sidebar ID), where the_widget is for only rendering a specific widget, usually only used for something that has no options that need to be edited.