How do I bring a page template into a new theme with separate styling?

The problem is that your theme still load the main style.css sheet, because it is called by wp_head(). See the documentation here

So, in order to not load your theme main stylesheet, you have to add to your functions.php file :

add_action( 'wp_enqueue_scripts', 'yolo_my_custom_template' );
function yolo_my_custom_template(){
    if ( is_page_template( 'name_of_your_page_template.php' ) ) :
      wp_deregister_script('twentynineteen-style');
      wp_deregister_script('twentynineteen-print-style');
    endif;
}