Sometimes adding these sites manually can be tricky. I also think the documentation needs to be updated for this. On my end, for Youtube, the search string is: https://www.youtube.com/results?search_query=%s&page={startPage?}&utm_source=opensearch
I would recommend using the OpenSearch indexing to add custom search engines to the browser. Just go to brave://settings/search and ensure that the Index other Search engines option is enabled.
Then, visit any site with a search engine (such as Youtube.com) and perform a search. You’ll then see that site appear in the Inactive shortcuts section of brave://settings/searchEngines. Click the Activate button and you should be good to go.