A more robust version of this code, and code from my original question, can be found here: https://gist.github.com/mhulse/5860561

class.foo_widget.php

<?php

class Foo_Widget extends WP_Widget
{
	
	public function __construct($arr = array()) { # If I don't set a default array() value, I get this notice: "Undefined variable: arr in /.../class.foo_widget.php on line 9"
		
		$args = wp_parse_args(
			$arr,
			array(
				'id_base' => '',
				'name' => '',
			)
		);
		
		extract($args, EXTR_SKIP);
		
		$widget_ops = array('classname' => __CLASS__, 'description' => __('Description goes here.'));
		$control_ops = array('width' => 400, 'height' => 350);
		
		# THIS WORKS:
		//parent::__construct('eeeeeeeeee', 'wtf son?', $widget_ops, $control_ops);
		
		# `$id_base` and `$name` are not getting passed:
		parent::__construct($id_base, $name, $widget_ops, $control_ops); # Why doesn't this work? Is it a timing thing?
		
		add_action(
			'widgets_init',
			function() {
				register_widget(__CLASS__);
			}
		);
		
	}
	
	public function update($new_instance, $old_instance) {
		
		$instance = $old_instance;
		
		$instance['text'] = $new_instance['text']; // Unfiltered HTML.
		
		return $instance;
		
	}
	
	public function form($instance) {
		
		$instance = wp_parse_args((array) $instance, array('text' => '',));
		
		$text = esc_textarea($instance['text']);
		
		?>
		
		<textarea rows="16" cols="20" name="<?=$this->get_field_name('text')?>" id="<?=$this->get_field_id('text')?>" class="widefat"><?=$text?></textarea>
		
		<?php
		
	}
	
	public function widget($args, $instance) {
		
		extract($args);
		
		$text = apply_filters(
			'widget_text',
			(empty($instance['text']) ? '' : $instance['text']),
			$instance
		);
		
		echo $text;
		
	}
	
}

functions.php

<?php

include_once('class.foo_widget.php');

$foo_test_widget = new Foo_Widget(
	array(
		'id_base' => 'foo_test_widget',
		'name' => 'Foo Test Widget'
	)
);

So, it looks like I should just stick to the more traditional way of creating custom widgets?

Yes, I think unfortunately you'll have to stick with the traditional methods.

FYI to future readers, the wp-hackers discussion can be found here: http://wordpress-hackers.1065353.n5.nabble.com/Need-WP-PHP-guru-eyeballs-feedback-on-my-WP-Widget-wrapper-class-td41674.html

Yes, I think unfortunately you'll have to stick with the traditional methods.

@JDGrimes sounds good. I don't mind the traditional methods ... I just thought it might be cool to setup something similar for widgets like WPAlchemy script does for meta boxes.

Honestly, more than anything, I'm glad to know that the results I'm getting are not just a limitation of my PHP knowledge. 😄

Ok, moving on now. Thank you! 👍

For those interested, this is my "traditional" sidebar widget setup:

https://gist.github.com/mhulse/5864117