WC products expire BUT still visible in shop and search results

Hi,
I’m using this plugin since ages and I’ve always been really happy.
I use post expirator with Woocommerce to hide date based products (classes) after a certain date.

I just saw that if I set a cross/upsell product, this will be shown even if already expired (and set to draft).
I had a look at the products back-office page and realized that a product can be in draft status, but still visible in the WC catalog (Catalog visibility: Shop and search results).

It would be nice having the option to change the Catalog visibility: to “hidden”, this would really expire and hide the products custom post type, both for WordPress AND for Woocommerce.

If there’s already this option it would me good to know, I looked everywhere but couldn’t find a way to solve this issue. Tank you.

All the best