Adding post fields in wp-json/wp/v2/search

I am trying to add the post (pages, posts) excerpt in the wp-json/wp/v2/search get endpoint but this endpoint seems to not take into account the register_rest_field method.

This causes wp-json/wp/v2/pages and wp-json/wp/v2/posts to die, but wp-json/wp/v2/search works perfectly, although it renders posts and pages.

Any idea how I can add my excerpt to the results of the wp-json/wp/v2/search route?