Can’t wrap my head around this function – any help would be appreciated

Your problem is that you return nothing if some of the options doesn’t meet your criteria. For example, you are doing this:

if ( $header_container == get_theme_mod( 'header_container_background_color', '#e6e6e6' ) ) 
    return;

So, if $header_container is equal to #e6e6e6, the function return and stop further execution. You have to change that logic. For example (just a quick example):

function wpforge_customize_css() {

    $output="";

    $header_container = esc_attr(get_theme_mod( 'header_container_background_color' )); 
    if ( $header_container != get_theme_mod( 'header_container_background_color', '#e6e6e6' ) ) {
        $output=".header_container{background-color: ".$header_container.'}';
    }

    $header_wrap_width = esc_attr(get_theme_mod( 'header_width' )); 
    if ( $header_wrap_width != get_theme_mod( 'header_width', '64rem' ) ) {
        $output=".header_wrap{background-color: ".$header_wrap_width.'}';
    }

    if( !empty($output) ) {

    ?>
        <!--Customizer CSS-->
        <style type="text/css" id="wpforge-customizer-css">
           <?php echo $output; ?>
        </style>
        <!--/Customizer CSS-->
    <?php
    }
}
?>