Okay, just so you know, don’t look in the archive-product.php and taxonomy… files.
The header image and page title area in this theme is handled by the theme’s header.php file. Look at the end of that file and you will see reference to the trusted_header_title() and trusted_custom_header_title() functions. The custom header setting for a particular page, category, tag etc. determines which of these two functions is to run.
It may be that you can use conditional logic to tell all product archives to run an empty custom header. Check out WooCommerce’s conditional tags for the appropriate one(s) to use.