Have custom post type Visibility be private by default, and have radio button also set to “Private”

Actually all you have to do is changing the script part to this:

<script type="text/javascript">
    (function($){
        try {
            $('#post-visibility-display').text('<?php echo $visibility_trans; ?>');
            $('#hidden-post-visibility').val('<?php echo $visibility; ?>');
            $('#visibility-radio-<?php echo $visibility; ?>').attr('checked', true);
        } catch(err){}
    }) (jQuery);
</script>

You see there is an additional line compared to your code. I’m referring you to my original answer for more information, this is element of the second – 2. – part of the answer.