You are correct sir! Sorry about that. I have fixed it and uploaded a new file. I changed $modules to $return.

Yes, it's working ... at first sight.
But there's something really wrong that will cause bad side effects:
1.
Your original function (you posted here) reads:

function com_intro_widget_hook($hook, $type, $return, $params) {

and this one is correctly returning

return $return;

while in your component the function reads:

function com_intro_widget_hook($hook, $type, $modules, $params) {

but you're still using the former code that is returning $return.

Here's the new component Y'all. Enjoy!

https://www.opensource-socialnetwork.org/component/view/7001/intro

Boom! It works!!! Had to pull out a few lines of exorbitant code and make it look a bit more like the first function to get it to go. Will have it posted as a new component shortly!

Yes.
In the beginning you got the original intro data via $params['user']->intro
Next, you passed it to textarea's purifier which returned it as $intro_content
Next, you passed $intro_content to textarea's responsifier which returned it as $intro_content
Thus $intro_content indeed is the finalized intro data ready for being passed to your profile widget.

Ok so, should it be $intro_content instead?

Name it as you want, at least with your current code it makes no sense because you don't pass it to the profile/widget plugin. You're passing the $display array instead.

So let's have a closer look into that.

$display['content'] = $content;

I can't find any initialization of $content in your function and I'm sure there's a PHP warning like ""Undefined variable $content" in file ...." in your error log.

It was an appeal to redundancy. The question, how do we make the function port like the first function? I was just trying to replicate it.

Okay, so let's have a look into your com_intro_widget_hook function:
Being called, it retrieves a complete User object with its attributes via $params['user']
And if all conditions are met the original $params['user']->intro data is passed to two post-processing hooks and in the end the processed intro data is stored in $intro_content.
Things are getting strange with the next line, though:

$params['user']->intro = $intro_content;

Why are you overwriting the original data with the processed data? Even more since $params['user'] is no longer used in the following lines?

Fairly sure the issue lies here...
The first function ports the Intro into a subpage (works great).
The second function should port the Intro into a widget (works some).
Any thoughts?

    function com_intro_profile_intro_page($hook, $type, $return, $params)
{
    $page = $params['subpage'];
    if ($page == 'intro' && isset($params['user']->intro)) {
        if (!isset($params['user']->intro_access) || (isset($params['user']->intro_access) && $params['user']->intro_access == OSSN_PUBLIC || ($params['user']->intro_access == OSSN_FRIENDS && ossn_isLoggedin() && ossn_loggedin_user()->isFriend(ossn_loggedin_user()->guid, $params['user']->guid)) || (ossn_isLoggedin() && ossn_loggedin_user()->guid == $params['user']->guid) || ossn_isAdminLoggedin())     ) {
            $intro_content = ossn_call_hook('textarea', 'purify', false, $params['user']->intro);
            $intro_content = ossn_call_hook('textarea', 'responsify', false, $intro_content); 
            $params['user']->intro = $intro_content;
            $content = ossn_plugin_view('profile/intro', $params);
            echo ossn_set_page_layout('module', array(
                'title' => ossn_print('com:intro:pagetitle'),
                'content' => $content
            ));
        }
    }
}
function com_intro_widget_hook($hook, $type, $return, $params) {
    if (!isset($params['user']->intro_access) || (isset($params['user']->intro_access) && $params['user']->intro_access == OSSN_PUBLIC || ($params['user']->intro_access == OSSN_FRIENDS && ossn_isLoggedin() && ossn_loggedin_user()->isFriend(ossn_loggedin_user()->guid, $params['user']->guid)) || (ossn_isLoggedin() && ossn_loggedin_user()->guid == $params['user']->guid) || ossn_isAdminLoggedin())     ) {
        $intro_content = ossn_call_hook('textarea', 'purify', false, $params['user']->intro);
        $intro_content = ossn_call_hook('textarea', 'responsify', false, $intro_content); 
        $params['user']->intro = $intro_content;
        $display = array();
        $display['content'] = $content;
        $content = ossn_plugin_view('profile/widget', $display);
        $return[] = $content;
    }
    return $return;
}