At brave://settings/search/privateSearch, originally the normal and private windows would both use the Brave search engine. After changing the configuration of search engines to use Startpage by default, normal windows will use Startpage for the search engine, but nothing is specified for private windows. Clicking the Change button does nothing (no change to the screen, no list from which to select).
As a test, I opened a private window, and entered a string in the address bar. Startpage was used for the search which is my preference.
Is the search engine listed for private windows left blank if the same one is used for normal windows? What if I wanted a different search engine in private windows. The Change button does nothing.