Search delisting and website removal solve different parts of the piracy problem.
Search delisting
An eligible copyright request can ask a search engine to stop showing a specific infringing URL in relevant search results. The original website may still remain online.
Website removal
Removing the source means acting against the page, host, platform, file provider or another service that can affect the original copy.
Why both matter
When both routes are available, source action can reduce access while search action can reduce discovery.