I just merged a new feature:
https://github.com/discourse/discourse/pull/24428
This adds a new search_default_sort_order site setting, set to “relevance” by default, that controls the default sort order for the full page /search route.
If the user changes the order in the dropdown on that page, we remember their preference automatically, and it takes precedence over the site setting as a default from then on. The preference is stored in browser LocalStorage, so it is device-specific.