No problem at all, and you are correct.
Basically, the search plugin cannot make direct changes to the results page, but it sort-of passes it’s own results via an API to the WooCommerce/Regular search handlers (whichever page is chosen on the search back-end). Then it is up to the results page handler to do the view/layout process.
Most of the time WooCommerce results pages does not support displaying other than ‘product’ custom post types. If anything else than a product is detected, it is usually removed by the WooCommerce handler process.