Solution:
You should reach out to your hosting provider asking if “bots redirecting” feature is activated for your site. This has been a confirmed cause of WooCommerce checkout page stuck loading its contents on several sites I reviewed.
The actual problem is that ajax call to /?wc-ajax=update_order_review on checkout, instead of reaching and being processed by WooCommerce backend, is redirected to the site homepage by the server with “bots redirecting” turned on.
Disabling “bots redirecting” should result in the issue being resolved. Here are instructions on how to do it on WP Engine hosting.