Ability to set products to not show on site

Proposing an update to Catalog Organization.

Make it so products that aren't assigned to a category are not visible on the site (even when searched). As it is now, if you remove a product from all categories in the catalog organization, it is invisible to users when navigating their site. However, if they search for that product, it appears and can be ordered. Creating a strange instance where there are phantom products that can't be seen unless searched for.

We have a few sites that do monthly and quarterly promos and have 100s of products that may or may not be used in a given promo. So I add and remove them from the catalog organization as needed. However, I just found out this weird bug on the site that allows them to order products that aren't in any visible category.

I can't think of any instance where a product that isn't assigned to any category, and therefore not visible on any part of the site would still be findable when searched.