How can I get a script to run every time the WooCommerce cart is empty?

I think you may be missing the AJAX functionality that happens behind the scenes when you remove an item. There’s a difference between running your code at page load vs. what runs after you remove some products and eventually end up with an empty cart. This explains what you’re noticing with it working upon page refresh.

In addition to your code that you’re already using, you may also need something like this:

function my_cart_updated( $item_id ) {

    // check if cart contents are empty
    if ( WC()->cart->get_cart_contents_count() == 0 ) {
        
        // do whatever you want here
        // ...
    }
}

// add the action
add_action( 'woocommerce_cart_item_removed', 'my_cart_updated' );

I think something like this may work:

function my_cart_updated( $item_id ) {

    // check if cart contents are empty
    if ( WC()->cart->get_cart_contents_count() == 0 ) {
    ?>

      <script type="text/javascript" defer>
          let cartTitle = document.querySelector('.entry-title');
          let progressBar = document.querySelector('.pbar-cart');
          let customHeading = document.querySelector('.empty-heading');
          let customDescription = document.querySelector('.empty-description');
          let customTruck = document.querySelector('.empty-truck');
          let cartCategories = customTruck.nextElementSibling;
          let standardNotice = customTruck.nextElementSibling.nextElementSibling;

          cartTitle.style.display = "none";
          progressBar.style.display = "none";
          standardNotice.style.display = "none";
          customHeading.style.display = "block";
          customDescription.style.display = "block";
          customTruck.style.display = "block";
          cartCategories.style.display = "block";
      </script>

    <?php
    }
}

// add the action
add_action( 'woocommerce_cart_item_removed', 'my_cart_updated' );