missing emails and outdated templates

I’ve been asked to solve a few problems on this website. Initially some pages were not showing, so I updated some of the plugins and that problem was resolved. However no emails, either from feedback forms or from the WooCommerce cart ever get to their recipients. Because this was 2 different plugins I figured it was a WordPress fault. I installed WP Mail SMTP and a couple of others, but nothing resolved it. I can see the sent messages in the entries tab for Gravity Forms (I also installed Fluent Forms just in case) and also can see the orders in the Orders tab for Woocommerce but the emails don’t arrive (not to the destination nor to a temporary address outside the domain).
So I cloned the site using UpdraftPlus so I could play with the plugins. I disabled most plugins and caused a site failure, so I activated WooCommerce and it came back. I upgraded this cloned site to WooCommerce 7.3.0 and that showed up some error screens and now the orders look like they get processed, but don’t show in the Orders tab as well as not send the email.
This site was originally developed probably more than 10 years ago …. so here is my problem!
Currently running PHP 7.3.33
Wordress updated to 6.1.1
WooCommerce version 4.6.5 – on cloned site upgraded to 7.3.0 but is problematic
Gravity Forms version 2.6.9

Going to the Woocommerce status page – I have more than 100 pages of overrides which are outdated. eg

/woocommerce/archive-product.php version 2.0.0 is out of date. The core version is 3.4.0,
dunbar-services/woocommerce/cart/cart-empty.php version 3.1.0 is out of date. The core version is 3.5.0,
dunbar-services/woocommerce/cart/cart.php version 3.1.0 is out of date. The core version is 3.8.0,
dunbar-services/woocommerce/cart/cross-sells.php version 3.0.0 is out of date. The core version is 4.4.0,
dunbar-services/woocommerce/cart/mini-cart.php version 3.1.0 is out of date. The core version is 3.7.0,

I’m sorry this question is so long. Are these out-of-date templates likely to be causing these errors? Bearing in mind that the client wants to make major changes to the site very shortly, is there a quick fix to get the emails flowing again while the decision is made on what changes are required?

The page I need help with: [log in to see the link]